Neutrosophic Precalculus and Neutrosophic Calculus

‫‪Neutrosophic Science International Association‬‬

‫يثادئ انرفاضم وانركايم انُ​ُىذروصىفكً‬ ‫و‬ ‫حضاب انرفاضم وانركايم انُ​ُىذروصىفكً‬ ‫ذأنُف‬ ‫االصرار انذكرىر فهىرَرٍ صًاراَذاكح‬ ‫ذرجًح‬ ‫انذكرىرج هذي اصًاػُم خانذ اصًاػُم انجًُهٍ‬ ‫انًهُذس احًذ خضر ػُضً انجثىرٌ‬ ‫لضى انرَاضُاخ‪ -‬كهُح انررتُح االصاصُح – جايؼح ذهؼفر ‪ -‬انؼراق‬

‫يصال نُظرَح انمًُح انىصطً انُ​ُرروصىفكُح‬

‫‪Neutrosophic Science International Association‬‬

‫‪Pons asbl‬‬ ‫‪5, Quai du Batelage, Brussells, Belgium, European Union‬‬ ‫‪ISBN : 978-1-59973-497-2‬‬

‫‪© Translators, author‬‬ ‫‪and the publisher, 2016.‬‬

‫يراجؼح انررجًح‬ ‫‪ -1‬األصرار انذكرىر أحًذ ػثذ انخانك صاليح‬ ‫رئُش لضى انرَاضُاخ وػهىو انحاصة تكهُح انؼهىو – جايؼح تىرصؼُذ – يصر‬

‫‪ -2‬انذكرىرج هىَذا ػثذ انحًُذ انغىانثٍ‬ ‫لضى انفُزَما وانرَاضُاخ انهُذصُح – كهُح انهُذصح – جايؼح تىر صؼُذ – يصر‬

‫‪ -3‬االصرار صؼُذ ترويٍ‬ ‫يخرثر يؼانجح انًؼهىياخ‪ ،‬كهُح انؼهىو تٍ ‪ ،M'Sik‬جايؼح انحضٍ انصاٍَ‪،‬‬ ‫‪ ،B.P 7955‬صُذٌ ػصًاٌ‪ ،‬انذار انثُضاء‪ ،‬انًغرب‬ ‫‪ -4‬و‪.‬و‪ .‬ػثذانًُؼى ػثذهللا خهف انذنًٍُ‬ ‫يراجؼح نغىَح‬ ‫لضى انهغح انؼرتُح ‪ -‬كهُح انررتُح االصاصُح – جايؼح ذهؼفر ‪ -‬انؼراق‬

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‫‪Neutrosophic Science International Association‬‬

‫ذمذَى انكراب‬ ‫إن التعبيررررعبن بب عيرررري االمرررر ليي ال رررري ب لبعوفيموعمرررر عل امي و ل رررر‬ ‫الج ي ‪,‬وفمعة التعج ي ل متوعة ه ى إمر بي ا لر وال دلر ح د ر اضرع بيمر‬ ‫ن الععاق الش يق وبن عم لتدم في التعج ي أل ت بي ت ال ل ق الج ي ( م ب‬ ‫التف ضررر والتم ررر الليتعومررروفمي ل رررو ردررروع إلررر اللوعو ر هعبر تررري في ررر‬ ‫وضررع ف رروعلتن ررن بلرر ا ال ل ررق الج يرر ا رر ع الع يرر ررن التمرر ع ت فرري ج يرر‬ ‫ال ج ت ووض بص ي في ب و الع ا والب ين ن عامتدم لدذا الفمرع الج ير‬ ‫والمتب ‪ ,‬والتي دلشأ فيد ل ج ير ا وهرو ر يمر بر ل ل ق‬ ‫‪ ,‬ن اال األب‬ ‫الليتعومرروفيو وإ ت ر ا لدررذا وض ر فعيررق الع ر ررن ج ر عتي ( بوعمررعي ‪ -‬صررع‬ ‫وت عفررع‪ -‬العررعاق ب ش ر عمي م ر عل امي تررم وض ر دمررح لع رروم ج ي ر ة فرري ج ر ت‬ ‫العي ضرري ت وا ص ر ا وب رروم ال مررب ولرررم ال ع و ر ت وإ ت ر ذلررو لوض ر اليرري‬ ‫ج ي ر ة لت ي ر الرررواهع الموليرري وال جت عيرري وي ترروه هررذا المت ر ب ب ر ‪ 34‬إش ر عة‬ ‫ن ج عتي بوعمرعي ‪ ,‬صرع‪ ,‬وت عفرع‪ ,‬العرعاق‪ ,‬و ر‬ ‫فعيق الع‬ ‫عجعيي ألب‬ ‫الجدررر ال ضرررلي الرررذه بذلررر ال تعج رررون فررري تعج ررري هررر ذا المتررر ب الضرررام تعررر‬ ‫ال واضي في ج التف ض والتم ر مأ ر الت بي ر ت الج ير ة ل ليتعومروفيو الرذه‬ ‫يعتبع دو عج ب ل غي الععبيي في ت بي ت الليتعوموفيو ‪.‬‬ ‫‪ٝ‬هجَ اُخزبّ كإنٗ‪ ٢‬و‪ ٝ‬وٕ وروإمّ ثخإبُا اُرإٌ ‪ٝ‬اُزوإم‪ ٣‬لُإ‪ً ٠‬إَ ٓإٖ صإب‪ ْٛ‬كإ‪٢‬‬ ‫ٓر ‪ٝ‬ع ر عٔخ ‪ٛ‬إاا اٌُزإبة‪ًٔ .‬إب وروإمّ ثخإبُا اُرإٌ ‪ٝ‬اُزوإم‪ٌُ ٣‬إعَ ٓإٖ صبػإعم كإ‪٢‬‬ ‫ٓ اعؼخ اُز عٔخ ‪ ٝ‬لخ اط ‪ٛ‬اا اٌُزبة لُ‪ ٠‬اُ٘إ‪ًٔ . ٞ‬إب وصإ ٍ ا اُؼِإ‪ ٢‬اُوإم‪ ٣‬وٕ ‪٘٣‬لإغ‬ ‫ث‪ ٚ‬غبُج‪ ٢‬اُؼِْ ‪ٝ‬أُؼ كخ ك‪ٝ ٢‬غ٘​٘ب اُؼ ث‪ ٢‬ػِ‪ ٠‬عٔ‪٤‬غ ٓضز‪٣ٞ‬بر‪ ْٜ‬األًب ‪٤ٔ٣‬إخ ‪ٝ‬اُؼِٔ‪٤‬إخ‬ ‫‪ٝ‬اُضوبك‪٤‬خ‪.‬‬ ‫‪ٝ‬ا ٖٓ ‪ ٝ‬اء اُوصم‪،،،‬‬ ‫أ‪.‬د‪.‬أحمد عبد الخالق سالمة‬ ‫قسم الرياضيات وعلوم الحاسب ‪ -‬كلية العلوم – جامعة بورسعيد – مصر‬ ‫‪6102‬‬

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‫‪Neutrosophic Science International Association‬‬

‫انًحرىَاخ‬ ‫خ‬ ‫‪1‬‬ ‫‪1.1‬‬ ‫‪1.1‬‬ ‫‪1.1‬‬ ‫‪4.1‬‬ ‫‪1.1‬‬ ‫‪1‬‬ ‫‪1.1‬‬ ‫‪1.1‬‬ ‫‪1.1‬‬ ‫‪4.1‬‬ ‫‪1.1‬‬ ‫‪6.1‬‬ ‫‪1.1‬‬ ‫‪2.1‬‬ ‫‪3.1‬‬ ‫‪12.1‬‬ ‫‪11.1‬‬ ‫‪11.1‬‬ ‫‪11.1‬‬ ‫‪14.1‬‬ ‫‪11.1‬‬ ‫‪16.1‬‬ ‫‪11.1‬‬ ‫‪12.1‬‬ ‫‪13.1‬‬ ‫‪12.1‬‬ ‫‪1‬‬ ‫‪1.1‬‬ ‫‪1.1‬‬ ‫‪1.1‬‬ ‫‪4.1‬‬ ‫‪1.1‬‬

‫انصفحح‬ ‫اصى انًىضىع‬ ‫‪4‬‬ ‫انًحرىَاخ‬ ‫‪6‬‬ ‫يمذيح ػٍ انًؤنف (صُرج ػانى وكاذة وفُاٌ)‬ ‫‪11‬‬ ‫انرًهُذ (انًفهىو انُ​ُىذروصىفكٍ فٍ حضاب انرفاضم وانركايم )‬ ‫‪11‬‬ ‫ٗظ ح ػبٓخ‬ ‫‪11‬‬ ‫ٓومٓخ‬ ‫‪11‬‬ ‫اُل ‪ٝ‬م ث‪ ٖ٤‬رؾِ‪ َ٤‬اُلز ح ‪ٝ ،‬رؾِ‪ َ٤‬أُغٔ‪ٞ‬ػخ ‪ٝ ،‬اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫‪11‬‬ ‫أُوب‪٤٣‬ش اُ‪ٜ٘‬مص‪٤‬خ األ‪٤ُٝ‬خ ؿ‪ ٤‬أُؾم ح‬ ‫‪13‬‬ ‫ه‪ٞ‬اٗ‪ ٖ٤‬ك‪٤‬ز‪٣‬بئ‪٤‬خ ؿ‪ٓ ٤‬ؾم ح‬ ‫‪12‬‬ ‫انحضاب انرًهُذٌ نهرفاضم وانركايم انُ​ُىذروصىفكٍ‬ ‫‪12‬‬ ‫اُؼِٔ‪٤‬بد اُغج ‪٣‬خ ُِٔغبٓ‪٤‬غ‬ ‫‪12‬‬ ‫اُؼالهبد ث‪ ٖ٤‬أُغبٓ‪٤‬غ اُغزئ‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪11‬‬ ‫اُخ أُغٔ‪ٞ‬ػخ اُغزئ‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪11‬‬ ‫اُماُخ اُ‪ٜ‬رخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪11‬‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُؼبٓخ‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُ‪ٜ‬رخ ا‪ ١‬اُز‪ٓ ٢‬غبُ‪ٜ‬ب ‪ٓٝ‬غبُ‪ٜ‬ب أُوبثَ ٓغبٓ‪٤‬غ ‪11‬‬ ‫عزئ‪٤‬خ‬ ‫‪12‬‬ ‫اُالرؾم‪٣‬م أُزوطغ ‪ٝ‬ؿ‪ ٤‬أُزوطغ‬ ‫‪12‬‬ ‫‪ٝ‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثو‪ٓ ْ٤‬زغ‪ٜ‬خ ذاد ٓزـ‪ ٤‬اد ٓزؼم ح‬ ‫‪13‬‬ ‫اُم‪ٝ‬اٍ اُعٔ٘‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪13‬‬ ‫ر ً‪٤‬ت اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪12‬‬ ‫ٓؼٌ‪ٞ‬س اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪14‬‬ ‫وصلب اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪14‬‬ ‫الرؾم‪٣‬ماد اُماُخ‬ ‫‪14‬‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُز‪ٝ‬ع‪٤‬خ‬ ‫‪11‬‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُل ‪٣‬خ‬ ‫‪16‬‬ ‫اُ٘ٔ‪ٞ‬ذط اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫‪11‬‬ ‫ٓؼبَٓ اال رجبغ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫‪12‬‬ ‫اُماُخ األص‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪13‬‬ ‫اُمُخ اُِ‪ٞ‬ؿب ‪٣‬زٔ‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪41‬‬ ‫ر ً‪٤‬ت اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪41‬‬ ‫حضاب انرفاضم وانركايم انُ​ُىذروصىفكٍ‬ ‫‪41‬‬ ‫اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪44‬‬ ‫ٓالءٓخ ( اُجؼم – اُغزئ‪) ٢‬‬ ‫‪44‬‬ ‫خ‪ٞ‬اص ٓالءٓخ (اُجؼم – اُغزئ‪)٢‬‬ ‫‪41‬‬ ‫اُلعبء (أُز ‪ - ١‬اُغزئ‪)٢‬‬ ‫‪41‬‬ ‫‪ُِ ε‬ـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُ‪٤‬ض ‪ٟ‬‬ ‫رؼ ‪٣‬ق‬ ‫‪4‬‬

‫‪Neutrosophic Science International Association‬‬ ‫‪6.1‬‬ ‫‪1.1‬‬ ‫‪2.1‬‬ ‫‪3.1‬‬ ‫‪12.1‬‬ ‫‪11.1‬‬ ‫‪11.1‬‬ ‫‪11.1‬‬ ‫‪14.1‬‬ ‫‪11.1‬‬ ‫‪16.1‬‬ ‫‪11.1‬‬ ‫‪12.1‬‬ ‫‪13.1‬‬ ‫‪12.1‬‬ ‫‪11.1‬‬ ‫‪11.1‬‬ ‫‪11.1‬‬ ‫‪14.1‬‬ ‫‪4‬‬ ‫‪1‬‬ ‫‪1.1‬‬ ‫‪1.1‬‬ ‫‪1.1.1‬‬ ‫‪1.1.1‬‬ ‫‪1.1.1‬‬ ‫‪4.1.1‬‬

‫ٓضبٍ ُؾضبة اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ؽبُخ خبصخ ُؾضبة اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ؽضبة اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ رؾِ‪٤ِ٤‬ب‬ ‫اُؾضبة اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ُ ٢‬ـب‪٣‬بد اُم‪ٝ‬اٍ اٌُض ‪٣‬خ‬ ‫شج‪ ٚ‬االصزٔ ا ‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ أُضزٔ ح‬ ‫ٗظ ‪٣‬خ اُو‪ٔ٤‬خ اُ‪ٞ‬صط‪ ٠‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ٓضبٍ ؽ‪ٗ ٍٞ‬ظ ‪٣‬خ اُو‪ٔ٤‬خ اُ‪ٞ‬صط‪ ٠‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ٓضبٍ ؽ‪ٗ ٍٞ‬ظ ‪٣‬خ اُو‪ٔ٤‬خ اُ‪ٞ‬صط‪ ٠‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ أُ‪ٞ‬صؼخ‬ ‫خ‪ٞ‬اص شج‪ -ٚ‬االصزٔ ا ‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫خ‪ٞ‬اص االصزٔ ا ‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ُِـب‪٣‬بد اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُالٗ‪ٜ‬بئ‪٤‬خ‬ ‫رؼ ‪٣‬ق‬ ‫وٓضِخ ػٖ اُـب‪٣‬بد اُالٗ‪ٜ‬بئ‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓغبُ‪ٜ‬ب ‪ٓٝ‬غبُ‪ٜ‬ب أُوبثَ ٓغبٓ‪٤‬غ‬ ‫االشزوبم اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬ؿ‪ ٤‬أُؾم‬ ‫اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬أُؾم‬ ‫رؼ ‪٣‬ق ٓجضػ ُِزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬أُؾم‬ ‫رؼ ‪٣‬ق ػبّ ُِزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬أُؾم‬ ‫انفصم انراتغ ( انخاذًح )‬ ‫شثد انًصطهحاخ‬ ‫انفصم انخايش (انًصادر)‬ ‫أُصب اُؼ ث‪٤‬خ‬ ‫أُصب االٌِٗ‪٤‬ز‪٣‬خ‬ ‫ًزت ‪ٝ‬ثؾ‪ٞ‬س ٓ٘ر‪ ٞ‬ح‬ ‫ٓوبالد اظبك‪٤‬خ ؽ‪ ٍٞ‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ي‬ ‫ٓومٓبد ُٔؤرٔ اد ػبُٔ‪٤‬خ ‪ٝ‬ؽِوبد اص‪٤‬خ ؽ‪ ٍٞ‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ي‬ ‫وغب ‪٣‬ؼ ًز‪ ٞ‬ا‪ ٙ‬ك‪ ٢‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ي‬

‫‪46‬‬ ‫‪46‬‬ ‫‪42‬‬ ‫‪43‬‬ ‫‪12‬‬ ‫‪12‬‬ ‫‪12‬‬ ‫‪11‬‬ ‫‪11‬‬ ‫‪11‬‬ ‫‪16‬‬ ‫‪11‬‬ ‫‪12‬‬ ‫‪13‬‬ ‫‪62‬‬ ‫‪61‬‬ ‫‪64‬‬ ‫‪61‬‬ ‫‪61‬‬ ‫‪61‬‬ ‫‪62‬‬ ‫‪11‬‬ ‫‪11‬‬ ‫‪11‬‬ ‫‪11‬‬ ‫‪21‬‬ ‫‪122‬‬ ‫‪123‬‬

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‫‪Neutrosophic Science International Association‬‬

‫صُرج ػانى وكاذة وفُاٌ (يؤنف انكراب)‬ ‫االصرار انذكرىر فهىرَرٍ صًاراَذاكه ‪ /‬جايؼح َُى يكضُكى االيرَكُح‬

‫‪Personal pictures for Sir Florentin Smarandache‬‬ ‫‪(contradictory in handwriting( Vietnam -2016‬‬ ‫صىرج شخصُح نهضُذ فهىرَرٍ صًاراَذاكه ( َكرة تكهرا َذَه)‪ /‬فُرُاو – ‪2116‬‬

‫‪ُٝ‬م ‪ٛ‬اا اُؼبُْ ك‪ ٢‬اُؼبش ٖٓ ‪٣‬ضٔج ػبّ ‪ 1314‬ك‪ٓ ٢‬م‪٘٣‬خ ‪ Balcesi‬ك‪ٓٝ ٢‬ب ‪ /‬ا‪٣‬طبُ‪٤‬ب‪ٞٛ ،‬‬ ‫ذُي اُؼبُْ أُ‪ٞ‬ص‪ٞ‬ػ‪ ٢‬اُا‪ ١‬ػَٔ ٓؤُلب‪ٓٝ ،‬ز عٔب‪ٓٝ ،‬ؾ ا ألًض ٖٓ ‪ً 122‬زبة ‪ٝ ،‬ثؾش ‪،‬‬ ‫‪ٓٝ‬وبُخ ػِٔ‪٤‬خ‪.‬‬ ‫اٗ‪ ٚ‬عَ ‪٣‬مػ‪ُِٜ٘ ٞ‬عخ ألٗ‪ٗ ٚ‬ر ك‪ ٢‬اُؼم‪٣‬م ٖٓ أُغبالد ‪ٝ‬اُؾو‪ ٍٞ‬اُؼِٔ‪٤‬خ‪ ،‬ػِ‪ ٠‬صج‪ َ٤‬أُضبٍ ال‬ ‫اُؾص ٗغم اٗ‪ ٚ‬هم اثمع ك‪ ٢‬انرَاضُاخ (ٗظ ‪٣‬خ األػما ‪ ، ،‬االؽصبء ‪ ،‬اُج٘‪ ٠‬اُغج ‪٣‬خ ‪ ،‬اُ‪ٜ٘‬مصخ‬ ‫اُاللهِ‪٤‬م‪٣‬خ ‪ٝ ،‬اُ‪ٜ٘‬مصخ اُضٔب اٗماً‪٤‬خ )‪ ،‬وػهىو انكًثُىذر (اُاًبء االصط٘بػ‪ٝ ،٢‬االٗرطب‬ ‫أُؼِ‪ٓٞ‬بر‪ ،)٢‬انفُزَاء (ك‪٤‬ز‪٣‬بء اٌُْ‪ ،‬ك‪٤‬ز‪٣‬بء اُغض‪ٔ٤‬بد)‪ ،‬االلرصاد (صوبكخ االهزصب ‪ٗ ،‬ظ ‪٣‬خ‬ ‫أُ اًز اُزغب ‪٣‬خ أُزؼم ح)‪ ،‬انفهضفح ( ُزؼٔ‪ ْ٤‬اُم‪٣‬بٌُز‪٤‬ي (اُغمٍ و‪ٓ ١‬وب ػخ اُؾغخ ثبُؾغخ)‬ ‫‪ٝ‬أُ٘طن اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ – ٢‬رؼٔ‪ُِ٘ٔ ْ٤‬طن اُعجبث‪ ٢‬اُؾمص‪ ،)٢‬انؼهىو االجرًاػُح (ٓوبالد ص‪٤‬بص‪٤‬خ)‬ ‫واألدب (اُرؼ ‪ٝ‬اُ٘ض ‪ٝ‬أُوبالد ‪ٝ‬اُ ‪ٝ‬ا‪٣‬خ‪ ،‬اُم آب‪ٓٝ ،‬ض ؽ‪٤‬بد األغلبٍ‪ٝ ،‬اُز عٔخ) وانفُىٌ‬ ‫(اُ صْ اُزغ ‪٣‬ج‪ /٢‬اُطِ‪٤‬ؼ‪ ،٢‬اُلٖ اُزص‪ ،١ ٣ٞ‬صْ ررٌ‪.)٢ِ٤‬‬ ‫‪٣ ٞٛٝ‬ؼَٔ ؽبُ‪٤‬ب وصزبذا ُِ ‪٣‬بظ‪٤‬بد ك‪ ٢‬عبٓؼخ ٗ‪ٌٓ ٞ٤‬ض‪ ٌٞ٤‬االٓ ‪٤ٌ٣‬خ‪ ٖٓٝ ،‬اٗغبزار‪ ٚ‬اُؼِٔ‪٤‬خ‬ ‫‪ٝ‬اُغ‪ٞ‬ائز اُز‪ ٢‬ؽصَ ػِ‪ٜ٤‬ب‪:‬‬ ‫‪6‬‬

‫‪Neutrosophic Science International Association‬‬ ‫‪-1‬‬

‫‪-1‬‬ ‫‪-1‬‬ ‫‪-4‬‬ ‫‪-1‬‬

‫ك‪ 11 ٢‬و‪،1211 ٍِٞ٣‬هبّ اُجبؽض‪ ٕٞ‬ك‪ ٢‬أُ٘ظٔخ اال‪ ٝ‬ث‪٤‬خ ُألثؾبس اُ٘‪٣ٝٞ‬خ (ص‪)ٕ ٤‬‬ ‫ثبإلصجبد اُغزئ‪ُ ٢‬ل ظ‪٤‬خ صٔب اٗماً‪ ٚ‬اُز‪ ٢‬ر٘ا ػِ‪ ٠‬اٗ‪ ٚ‬ال ‪ٞ٣‬عم ؽم اهص‪ُِ ٠‬ض ػخ ك‪٢‬‬ ‫اٌُ‪. ٕٞ‬‬ ‫ؽصَ ػِ‪ ٠‬عبئزح ٗ‪ٌٓ ٞ٤‬ض‪ ٌٞ٤‬ألكعَ ًزبة ػبّ ‪ٝ 1211‬ذُي ػٖ ًزبث‪" ٚ‬ث٘‪ ٠‬عج ‪٣‬خ‬ ‫عم‪٣‬مح " ٓ٘بصلخ ٓغ اُمًز‪ ٞ‬ح كبصبٗضب ًبٗماصبٓ‪.٢‬‬ ‫ؽصَ ػِ‪ ٠‬ش‪ٜ‬ب ر‪ً ٢‬ز‪ ٞ‬ا‪ ٙ‬كخ ‪٣‬خ ك‪ ٢‬ػبّ ‪ ٖٓ ًَ ٖٓ 1211‬ثٌ‪( ٖ٤‬عبٓؼخ ع‪٤‬ب‪ٝ‬ر‪ٗٞ‬ؾ‬ ‫)‪ ٖٓٝ ،‬ث‪ٞ‬خب صذ ( وًب ‪٤ٔ٣‬خ اً‪ٓٝ ٞ‬ب) ‪.‬‬ ‫ؽصَ ػِ‪ ٠‬اُ‪ٞ‬صبّ اُا‪ٛ‬ج‪ٓ ٖٓ ٢‬ؤصضخ ر‪٤ِ٤‬ض‪-ٞ٤‬ؿبُ‪ ٢ِ٤‬اُِ٘مٗ‪٤‬خ ُِؼِ‪ ّٞ‬ػبّ ‪ 1212‬لذ وه‪ْ٤‬‬ ‫ؽلَ اُزٌ ‪ ْ٣‬ك‪ ٢‬عبٓؼخ ث‪ٌ٤‬ش ‪٘ٛ ،‬ـب ‪٣‬ب‪.‬‬ ‫‪ ٞٛٝ‬و‪٣‬عب ػع‪ ٞ‬ك‪ ٢‬االًب ‪٤ٔ٣‬خ اُ ‪ٓٝ‬بٗ‪٤‬خ ‪ -‬األٓ ‪٤ٌ٣‬خ ُِؼِ‪.ّٞ‬‬ ‫‪٣‬ضزط‪٤‬غ اُوب ئ اٌُ ‪ ْ٣‬االغالع ػِ‪ً ٠‬زت اُض‪ ٤‬كِ‪ٗ ٞ‬زٖ ك‪ ٖٓ ًَ ٢‬أُ‪ٞ‬اهغ اُزبُ‪٤‬خ‪:‬‬ ‫)‪(Amazon Kindle, Amazon.com, Google Book Search‬‬

‫‪ٝ‬ك‪ ٢‬اُؼم‪٣‬م ٖٓ أٌُزجبد ك‪ ٢‬عٔ‪٤‬غ وٗؾبء اُؼبُْ ٓ٘‪ٜ‬ب ٌٓزجخ اٌُ‪ٗٞ‬ـ س (اُؼبصٔخ ‪ٝ‬اش٘طٖ)‪ ،‬ا‪٣‬عب‬ ‫ك‪ ٢‬هبػمح اُج‪٤‬بٗبد اُؼِٔ‪٤‬خ اُم‪٤ُٝ‬خ ‪ ، arXiv.org‬أُما ح ٖٓ هجَ عبٓؼخ ً‪( Cornell َ٤ٗ ٞ‬‬ ‫)‪ .University‬لٕ اُض‪ ٤‬كِ‪ٗ ٞ‬زٖ ‪ٝ ٖٓ ٞٛ‬ظغ ٗظ ‪٣‬خ ‪٣‬ز د‪ -‬صٔب اٗماً‪(Dezert- ٚ‬‬ ‫)‪ Smarandache theory‬ك‪ٞٓ ٢‬ظ‪ٞ‬ع االٗرطب أُؼِ‪ٓٞ‬بر‪ ٞٛٝ ٢‬اؽم ٓ‪ٞ‬اظ‪٤‬غ اُ ‪٣‬بظ‪٤‬بد‬ ‫اُزطج‪٤‬و‪٤‬خ ‪ ،‬ع٘جب لُ‪ ٠‬ع٘ت ٓغ اُمًز‪ ٖٓ J. Dezert ٞ‬ك ٗضب ‪ٛ‬ا‪ ٙ‬اُ٘ظ ‪٣‬خ ٓؼ ‪ٝ‬كخ ‪٤ُٝ‬ب ألٗ‪ٜ‬ب هم‬ ‫رْ اصزخمآ‪ٜ‬ب ك‪ٓ ٢‬غبٍ اُ ‪ٝ‬ث‪ٞ‬ربد‪ ،‬اُطت‪ٝ ،‬اُؼِ‪ ّٞ‬اُؼضٌ ‪٣‬خ‪ٝ ،‬ػِْ اُزؾٌْ ا‪ُِٜٔٝ ،٢ُ٥‬زٔ‪ٖٓ ٖ٤‬‬ ‫ذ‪ ١ٝ‬االخزصبص ٗغم اٗ‪ ٚ‬ص٘‪٣ٞ‬ب ‪٘ٓٝ‬ا ػبّ ‪ 1221‬رزْ ػ‪ٞ‬ح اُض‪ ٤‬كِ‪ٗ ٞ‬زٖ ُزوم‪ٓ ْ٣‬ؾبظ اد‬ ‫‪ٝ‬و‪ ٝ‬ام ػِٔ‪٤‬خ ؽ‪ٞٓ ٍٞ‬ظ‪ٞ‬ع االٗرطب أُؼِ‪ٓٞ‬بر‪ ٢‬ك‪ٓ ٢‬ؤرٔ اد ‪٤ُٝ‬خ ٓ٘‪ٜ‬ب ك‪ ٢‬وصز اُ‪٤‬ب‬ ‫(‪ ،)1221‬اُض‪٣ٞ‬م (‪ ،)1224‬اُ‪ٞ‬ال‪٣‬بد أُزؾمح األٓ ‪٤ٌ٣‬خ (‪ ،)1221‬ل‪٣‬طبُ‪٤‬ب (‪ً٘ ،)1226‬ما‬ ‫(‪ ،)1221‬ؤُبٗ‪٤‬ب (‪،)1222‬ك‪ ٢‬لصجبٗ‪٤‬ب (‪ ،)1226‬ثِغ‪ٌ٤‬ب (‪ٝ ،)1221‬ك‪ ٢‬عبٓؼبد وخ ‪ٓ ٟ‬ضَ‬ ‫لٗم‪٤ٗٝ‬ض‪٤‬ب ػبّ ‪ُِٔ 1226‬ز‪٣‬م ‪ ٌٖٔ٣‬اُ ع‪ٞ‬ع ُِٔ‪ٞ‬هغ‬ ‫(‪ )http://fs.gallup.unm.edu//DSmT.htm‬لذ صْٔ ‪ٛ‬اا أُ‪ٞ‬هغ ‪٣ٝ‬و‪ ّٞ‬ػِ‪ ٠‬ا ا ر‪ٚ‬‬ ‫‪ٝ‬ص‪٤‬بٗز‪ ٚ‬اُض‪٤‬م كِ‪ٗ ٞ‬زٖ ث٘لض‪.ٚ‬‬ ‫ػ‪ًٔ ٢‬زٌِْ ث ػب‪٣‬خ ‪ًٝ‬بُخ ٗبصب ك‪ ٢‬ػبّ ‪ ٖٓٝ 1224‬هجَ ؽِق شٔبٍ االغِض‪ ٢‬ػبّ ‪،1221‬‬ ‫ٗر د ثؾ‪ٞ‬ص‪ ٚ‬ك‪ٝ ٢‬هبئغ ‪ٛ‬ا‪ ٙ‬أُؤرٔ اد‪ٝ .‬هم ص‪ٞ‬ة اُؼم‪٣‬م ٖٓ وغب ‪٣‬ؼ اُمًز‪ ٞ‬ا‪ ٙ‬ك‪ ٢‬عبٓؼبد ٓضَ‬ ‫ً٘ما‪ٝ ،‬ك ٗضب‪ٝ ،‬ل‪٣‬طبُ‪٤‬ب‪ٝ ،‬ل‪ ٣‬إ‪.‬‬ ‫ك‪ ٢‬اُج٘‪ ٠‬اُغج ‪٣‬خ اُضٔب اٗماً‪٤‬خ ٗغم ٓل اد عج ‪٣‬خ ٓ‪ٜٔ‬خ ٓضَ أُ‪٣ٞٗٞ‬ماد‪ ،‬وشجب‪ ٙ‬اُزٓ ‪ ،‬كعبء‬ ‫أُزغ‪ٜ‬بد‪ ،‬اُغج اُخط‪ٝ ،٢‬ؿ‪ٛ ٤‬ب ‪ٝ‬ؽبُ‪٤‬ب ‪٣‬زْ رم ‪٣‬ض‪ٜ‬ب ُِطالة ك‪ ٢‬أُؼ‪ٜ‬م اُ‪ٜ٘‬م‪ُِ ١‬زٌ٘‪ُٞٞ‬ع‪٤‬ب ك‪٢‬‬ ‫ع‪٘٤‬ب‪ ،١‬ربٓ‪ٗ َ٤‬ب ‪ ،ٝ‬اُ‪ٜ٘‬م‪ٓٝ ،‬ب زاُذ ‪٘ٛ‬بى وغب ‪٣‬ؼ ُِمًز‪ ٞ‬ا‪ ٙ‬رؾذ لش اف اُمًز‪ ٞ‬ح ( كبصبٗضب‬ ‫ًبٗماصبٓ‪ ، )٢‬اُز‪ ٢‬رؼم لؽم‪ ٟ‬أُرب ًبد ك‪ ٢‬اُؼم‪٣‬م ٖٓ اُم اصبد ُِج٘‪ ٠‬اُغج ‪٣‬خ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ (اٗظ اُ اثػ ‪.) http://fs.gallup.unm.edu//algebra.htm‬‬

‫‪7‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ٖٓ اػٔبُ‪ ٚ‬أُ ٓ‪ٞ‬هخ ك‪ ٢‬اُ ‪٣‬بظ‪٤‬بد وٗ‪ ٚ‬هبّ ثز ص‪٤‬ش ‪ٝ‬رط‪ ٣ٞ‬أُ٘طن اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ ، ٢‬أُغبٓ‪٤‬غ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ‪ ،‬االؽزٔبُ‪٤‬خ ‪ٝ‬االؽصبء اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ٝ ،٢‬اُز‪ ٢ٛ ٢‬رؼٔ‪ٔ٤‬بد ُِٔ٘طن اُعجبث‪٢‬‬ ‫‪ٝ‬أُ٘طن اُعجبث‪ ٢‬اُؾمص‪ُِٔٝ ،٢‬غبٓ‪٤‬غ اُعجبث‪٤‬خ (ٗخا ثبُاً أُغبٓ‪٤‬غ اُعجبث‪٤‬خ اُؾمص‪٤‬خ)‪.‬‬ ‫ُوم وخزب ‪ٛ‬اا اُؼبُْ رضٔ‪٤‬خ ٓ٘طوخ اُ ‪٣‬بظ‪٤‬بر‪ ٢‬اُغم‪٣‬م ث صْ ( انًُطك انُ​ُىذروصىفٍ) ‪ ،‬لذ وٕ‬ ‫وصَ ‪ٛ‬ا‪ ٙ‬أٌُِخ ‪٣‬ؼ‪ ٍ ٞ‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ب ‪ًِٔ ٢ٛٝ Neutro- sophy‬خ ٓؤُلخ ٖٓ ٓوطؼ‪ ،ٖ٤‬اال‪ٍٝ‬‬ ‫‪ Neutro‬ثبُالر‪٤٘٤‬خ ‪ٝ ،‬ثبُل ٗض‪٤‬خ رِلع ‪ ٢ٛٝ Neutre‬رؼ٘‪ٓ( ٢‬ؾب‪٣‬م ‪ . )Neutral‬أُوطغ اُضبٗ‪٢‬‬ ‫ٌُِِٔخ ‪ًِٔ ٢ٛٝ Sophia‬خ ‪ٗٞ٣‬بٗ‪٤‬خ رؼ٘‪ ( ٢‬ؽٌٔخ ‪ ٖٓٝ ،)Skill/ Wisdom‬صْ ‪٣‬صجؼ ٓؼ٘‪٠‬‬ ‫أٌُِخ ثٔغِٔ‪ٜ‬ب " ٓؼ كخ اُلٌ أُؾب‪٣‬م " ‪ُِٔ ( .‬ز‪٣‬م ػٖ ذُي وٗظ ًزبة اُلِضلخ اُؼ ث‪٤‬خ ٖٓ‬ ‫ٓ٘ظ‪ٞ٤ٗ ٞ‬ر ‪ٝ‬ص‪ٞ‬ك‪ /٢‬صالػ ػضٔبٕ ‪ ٝ‬كِ‪ٗ ٞ‬زٖ صٔب اٗماًخ)‪.‬‬ ‫‪ًٝ‬بٕ ٓزؾمصب ك‪ ٢‬عبٓؼخ ث‪ ٢ًِ ٤‬ػبّ ‪ 1221‬ك‪ٓ ٢‬ؤرٔ ٗظٔ‪ ٚ‬االصزبذ اُر‪ ٤ٜ‬اُمًز‪ُ ٞ‬طل‪ ٢‬زا ا‬ ‫وث‪ ٞ‬أُ٘طن اُعجبث‪ ٝ . ٢‬ػ‪ ٢‬و‪٣‬عب ك‪ ٢‬اُ‪ٜ٘‬م (‪ ،)1224‬اٗم‪٤ٗٝ‬ض‪٤‬ب (‪ٓ ،)1226‬ص (‪.)1221‬‬ ‫‪٘ٛٝ‬بى وغ ‪ٝ‬ؽز‪ً ٢‬ز‪ ٞ‬ا‪ ٙ‬ػ٘‪ٜٔ‬ب ك‪ ٢‬عبٓؼخ ‪ٝ‬ال‪٣‬خ ع‪ ٞ‬ع‪٤‬ب ك‪ ٢‬ورالٗزب‪ٝ ،‬ك‪ ٢‬عبٓؼخ ً‪٘٣ٞ‬زالٗم ك‪٢‬‬ ‫وصز اُ‪٤‬ب (اٗظ اُ اثػ ‪.)http://fs.gallup.unm.edu//neutrosophy.htm‬‬ ‫لٕ أُلب‪ ْ٤ٛ‬اُضٔب اٗماً‪٤‬خ ك‪ٗ ٢‬ظ ‪٣‬خ األػما ٓؼ ‪ٝ‬كخ ػبُٔ‪٤‬ب‪ٓ ،‬ضَ ٓزضِضالد صٔب اٗماً‪ٝ ،ٚ‬اٍ‬ ‫صٔب اٗماً‪ٝ ،ٚ‬ص‪ٞ‬اثذ صٔب اٗماً‪ٞٓ ٢ٛٝ ( ٚ‬ع‪ ٞ‬ح ك‪ ٢‬أُ‪ٞ‬هغ أُ ٓ‪ٞ‬م " ٓ‪ٞ‬ص‪ٞ‬ػخ ‪CRC‬‬ ‫ُِ ‪٣‬بظ‪٤‬بد"‪ ،‬كِ‪٣ ٞ‬ما ‪1332‬؛ وٗظ اُ اثػ (‪ .)http://mathworld.wolfram.com‬ر‪ٞ‬عم‬ ‫اُؼم‪٣‬م ٖٓ اُم‪ٝ‬اٍ اُضٔب اٗماً‪٤‬خ ك‪ً " ٢‬زبة ُ٘ظ ‪٣‬خ األػما "‪ٗ ،‬ر ك‪ ٢‬ا اُ٘ر أُ ٓ‪ٞ‬هخ‬ ‫‪ ،Springer-Verlag‬ػبّ ‪ً ٖٓٝ ،1226‬زج‪ ٚ‬اُو‪ٔ٤‬خ " االػما اال‪٤ُٝ‬خ ك‪ ٢‬أُ٘ظ‪ ٞ‬اُؾضبث‪"٢‬‬ ‫اُطجؼخ اُضبٗ‪٤‬خ ٗر د ك‪ٗ ٢‬لش ا اُ٘ر وٗلخ اُاً ُِؼبّ ‪. 1221‬‬ ‫ُالغالع ػِ‪ٓ ٠‬ؤُلبد ػِٔ‪٤‬خ وخ ‪ُِ ٟ‬مًز‪ ٞ‬كِ‪ٗ ٞ‬زٖ صٔب اٗماً‪ ٚ‬ص‪ٞ‬اء ك‪ٗ ٢‬ظ ‪٣‬خ األػما و‪ ٝ‬ك‪٢‬‬ ‫اُز‪ٞ‬اكو‪٤‬بد‪ٝ ،‬اُز‪ٗ ٢‬ر د ك‪ ٢‬عبٓؼخ ‪ Xi'an‬ك‪ ٢‬اُص‪ ٖٓ ٖ٤‬خالٍ أُغِخ اُم‪٤ُٝ‬خ‬ ‫"‪( " Scientia Magna‬اٗظ ػم ‪ٛ‬ب األخ‪ ٤‬ػِ‪ ٠‬اُ اثػ اُزبُ‪:٢‬‬ ‫‪) http://fs.gallup.unm.edu//ScientiaMagna4no3.pdf‬‬ ‫‪ٝ‬األًب ‪٤ٔ٣‬خ اُص‪٤٘٤‬خ ُِؼِ‪ ّٞ‬ك‪ ٢‬ثٌ‪" ، ٖ٤‬أُغِخ اُم‪٤ُٝ‬خ ُِ ‪٣‬بظ‪٤‬بد اُز‪ٞ‬اكو‪٤‬خ" (اٗظ ػم ‪ٛ‬ب األخ‪٤‬‬ ‫ك‪.) http://fs.gallup.unm.edu//IJMC-3-2008.pdf :٢‬‬ ‫ُوم رْ ك‪ ٢‬اُؼبّ ‪ 1331‬ر٘ظ‪ٓ ْ٤‬ؤرٔ ‪ ٢ُٝ‬ؽ‪ ٍٞ‬أُلب‪ ْ٤ٛ‬اُضٔب اٗماً‪ ٚ٤‬ك‪ٗ ٢‬ظ ‪٣‬خ االػما‬ ‫ثغبٓؼخ ً ا‪ٞ٣‬كب‪ٓٝ ،‬بٗ‪٤‬ب (ؽ‪٤‬ش رخ ط ٓ٘‪ٜ‬ب ك‪ ٢‬اصز‪ ٚ‬اُغبٓؼ‪٤‬خ اال‪٤ُٝ‬خ ‪ًٝ‬بٕ اال‪ ٍٝ‬ػِ‪ ٠‬كؼز‪ٚ‬‬ ‫ػبّ ‪ ( ،)1313‬وٗظ اُ اثػ‬ ‫‪.) http://fs.gallup.unm.edu/ProgramConf1SmNot.pdf‬‬ ‫لٕ اُؼم‪٣‬م ٖٓ ‪ٛ‬ا‪ ٙ‬أُؤرٔ اد رْ رص٘‪٤‬ل‪ٜ‬ب ٖٓ هجَ أُغِخ اُؼِٔ‪٤‬خ أُ ٓ‪ٞ‬هخ ‪" Notice of the‬‬ ‫"‪ ، American mathematical Society‬وٗظ ػِ‪ ٠‬صج‪ َ٤‬أُضبٍ ‪ٝ‬هبئغ أُؤرٔ اد اُم‪٤ُٝ‬خ‬ ‫ٓ٘ا ‪ 1222 -1221‬ػِ‪ ٠‬اُ اثػ اُزبُ‪:٢‬‬ ‫‪8‬‬

‫‪Neutrosophic Science International Association‬‬ ‫)‪(http://fs.gallup.unm.edu//ScientiaMagna4no1.pdf‬‬ ‫‪ٓ ٞٛٝ‬ؾ أُغِخ اُم‪٤ُٝ‬خ " ‪ٝ ،" Progress in Physics‬اُز‪ ٢‬رطجغ ‪ٝ‬رؾ ك‪ ٢‬عبٓؼخ ٗ‪ٞ٤‬‬ ‫ٌٓض‪ٓ ، UNM ٌٞ٤‬غ ٓضب‪ٝ ٖ٤٤ُٝ ٖ٤ٔٛ‬ع‪ٜ‬بد اػ‪٤‬خ رٔضِ‪ٜ‬ب ػمح ٓؼب‪ٛ‬م ُالثؾبس اُ٘‪٣ٝٞ‬خ ٖٓ‬ ‫عٔ‪٤‬غ وٗؾبء اُؼبُْ‪ ُ .‬ؤ‪٣‬خ لؽم‪ ٟ‬لصما ار‪ٜ‬ب وٗظ اُ اثػ‬ ‫) ‪. ( http://fs.gallup.unm.edu//PP-03-2008.pdf‬‬ ‫وٓب ك‪ ٢‬اُل‪٤‬ز‪٣‬بء هبّ ثص‪٤‬بؿخ ٓل‪ٜٓٞ‬ب عم‪٣‬ما ‪٣‬مػ‪ ٠‬اُالٓب ح "‪ٝ ،"unmatter‬وظ‪ ٜ‬ص‪٘٤‬ب ‪ٞ٣‬‬ ‫اُز٘بهعبد أٌُ‪٤ٓٞ‬خ ثبصزخماّ أُ٘طن اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٘ٓ ٞٛٝ( ٢‬طن ٓزؼم اُو‪ُ ) ْ٤‬ز‪ٞ‬ص‪٤‬غ‬ ‫اُلعبءاد اُل‪٤‬ز‪٣‬بئ‪٤‬خ‪ًٔ ،‬ب ‪ٝ‬صغ أُؼب الد اُزلبظِ‪٤‬خ اُل‪٤‬ز‪٣‬بئ‪٤‬خ ٖٓ اُص‪٤‬ؾ اُ ثبػ‪٤‬خ اُ‪ ٠‬ص‪٤‬ؾ‬ ‫ثبػ‪٤‬خ ص٘بئ‪٤‬خ‪.‬‬ ‫وٗظ اُ اثػ (‪. )http://fs.gallup.unm.edu//physics.htm‬‬ ‫ك‪ ٢‬االهزصب ًزت ٓغ ‪ Vector Christianto‬ؽ‪ ٍٞ‬اُضوبكخ االهزصب ‪٣‬خ ًجما‪ُِ َ٣‬جِمإ أُزخِلخ‪،‬‬ ‫‪ٝ‬اهز ػ ٗظ ‪٣‬خ أُ اًز اُزغب ‪٣‬خ أُزؼم ح‪ .‬وٗظ اُ اثػ‬ ‫(‪. )http://fs.gallup.unm.edu//economics.htm‬‬ ‫ك‪ ٢‬اُلِضلخ همّ ر اً‪٤‬ت ٖٓ ػمح وكٌب كِضل‪٤‬خ ٓز٘بهعخ ‪ٓٝ‬ما س كٌ ‪٣‬خ‪ٝ​ٝ ،‬صغ عمُ‪٤‬بد اُل‪ِ٤‬ض‪ٞ‬ف‬ ‫االُٔبٗ‪٤ٛ ٢‬ـَ لُ‪ ٠‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ب‪ٓ ٞٛٝ ،‬ب ‪٣‬ؼ٘‪ ٢‬رؾِ‪٤ُ َ٤‬ش كوػ األظما ‪ ٌُٖٝ‬أُ ًجبد أُؾب‪٣‬مح‬ ‫ث‪ٛ ٖ٤‬ا‪ ٙ‬االظما ‪ .‬وٗظ اُ اثػ‬ ‫(‪. )http://fs.gallup.unm.edu//neutrosophy.htm‬‬ ‫ك‪ ٢‬اال ة ‪٣‬ؼم ٓؤصضب ُٔم صخ أُلب هبد ٓب ‪٣‬ؼ٘‪ ٢‬اُؾ ًخ أُؼبص ح اُوبئٔخ ػِ‪ ٠‬االصزخماّ‬ ‫أُل غ ُِٔز٘بهعبد ك‪ ٢‬اُزخِ‪٤‬ن ‪ٝ‬اُز‪ٝ ٢‬ظغ اصض‪ٜ‬ب ػبّ ‪ 1322‬ك‪ٓٝ ٢‬بٗ‪٤‬ب‪ٗ ٝ .‬ر ‪٤ُٝ‬ب خٔضخ‬ ‫ٓوزطلبد و ث‪٤‬خ ‪٤ُٝ‬خ ػٖ أُلب هبد‪ُِٔ ،‬ز‪٣‬م وٗظ اُ اثػ‬ ‫(‪. )http://fs.gallup.unm.edu//a/Paradoxism.htm‬‬ ‫ك‪ٔ٤‬ب ‪٣‬زؼِن ثبألغ اُغم‪٣‬مح ُِٔلب هبد ٗالؽع اٗ‪ ٚ‬همّ‪:‬‬ ‫• وٗ‪ٞ‬اع عم‪٣‬مح ٖٓ اُرؼ ث شٌبٍ صبثزخ‪.‬‬ ‫• وٗ‪ٞ‬اع عم‪٣‬مح ٖٓ اُوصخ اُوص‪ ٤‬ح‪.‬‬ ‫• وٗ‪ٞ‬اع عم‪٣‬مح ٖٓ اُم آب‪.‬‬ ‫• ‪ٝ‬وٗ‪ٞ‬اع عم‪٣‬مح ٖٓ اُخ‪٤‬بٍ اُؼِٔ‪ ٢‬ك‪ ٢‬اُ٘ض ‪.‬‬ ‫‪ ٌٖٔ٣ٝ‬رؾٔ‪ً َ٤‬زت ؽ‪ٛ ٍٞ‬ا‪ ٙ‬أُ‪ٞ‬اظ‪٤‬غ ٖٓ أُ‪ٞ‬هغ اُزبُ‪: ٢‬‬ ‫‪. http://fs.gallup.unm.edu//eBooks-otherformats.htm‬‬ ‫‪9‬‬

‫‪Neutrosophic Science International Association‬‬ ‫‪ ُٚٝ‬رغب ة و ث‪٤‬خ ُـ‪٣ٞ‬خ ك‪ٓ ٢‬غِم ثؼ٘‪ٞ‬إ‪ٓ" :‬ؼغْ كِ‪ٗ ٞ‬ز‪ ُٚ ،)1222( " ٖ٤‬آب ٓ٘ب‪ٛ‬عخ‬ ‫ُِمًزبر‪٣ ٞ‬خ ثؼ٘‪ٞ‬إ "ثِم اُؾ‪ٞ٤‬اٗبد"‪ ٢ٛٝ ،‬آب صبٓزخ! ػ ظذ ك‪ ٢‬أُ‪ ٜ‬عبٕ اُم‪ُِٔ ٢ُٝ‬ض ػ‬ ‫اُطالث‪ ،٢‬ثبُما اُج‪٤‬عبء (أُـ ة)‪ ،‬ثزب ‪٣‬خ ‪ 11-21‬ا‪ٝ 1331 ،ٍِٞ٣‬رِو‪ٛ ٠‬اا اُؼَٔ عبئزح‬ ‫خبصخ ٖٓ ُغ٘خ اُزؾٌ‪ًٔ . ْ٤‬ب ‪ٝ‬ػ ض ‪ٛ‬اا اُؼَٔ ٓ ح وخ ‪ ٟ‬ك‪ ٢‬ؤُبٗ‪٤‬ب ثزب ‪٣‬خ ‪ 13‬صجزٔج‬ ‫‪ .1331‬وٗظ اُ اثػ ُجؼط وػٔبُ‪ ٚ‬أُض ؽ‪٤‬خ‬ ‫( ‪.) http://fs.gallup.unm.edu//a/theatre.htm‬‬ ‫رؼ‪ٜ‬م ثز‪ٞ‬ؽ‪٤‬م اُ٘ظ ‪٣‬بد ك‪ ٢‬اُلٖ وٗظ اُ اثػ‬ ‫(‪.) http://fs.gallup.unm.edu//a/oUTER-aRT.htm‬‬ ‫‪ٝ‬ر‪ٞ‬عم ك‪ ٢‬عبٓؼخ ‪ٝ‬ال‪٣‬خ و ‪٣‬ز‪ٗٝ‬ب‪ٌٓ ،‬زجخ ‪ٛ‬ب‪٣‬مٕ‪ ، ،‬عٔغ ًج‪ ٖٓ ٤‬اٌُزت ‪ٝ‬أُغالد ‪ٝ‬أُخط‪ٞ‬غبد‬ ‫‪ٝ‬اُ‪ٞ‬صبئن ‪ٝ‬األه اص أُمٓغخ ‪ٝ‬وه اص اُل‪٤‬م‪ ٞ٣‬اُ هٔ‪٤‬خ ‪ٝ‬وش غخ اُل‪٤‬م‪ ٞ٣‬ػٖ وػٔبُ‪ٓ ُٚٝ ،ٚ‬غٔ‪ٞ‬ػخ‬ ‫خبصخ وخ ‪ ٟ‬ك‪ ٢‬عبٓؼخ رٌضبس ك‪ ٢‬و‪ٝ‬صزٖ‪ ،‬و ش‪٤‬ق اُ ‪٣‬بظ‪٤‬بد األٓ ‪ ( ٢ٌ٣‬اخَ ٓ ًز اُزب ‪٣‬خ‬ ‫األٓ ‪ٞٓ .)٢ٌ٣‬هؼ‪ ٚ‬ػِ‪ ٠‬شجٌخ اإلٗز ٗذ‪:‬‬ ‫‪//http://fs.gallup.unm.edu‬‬ ‫ُ‪ٜ‬اا أُ‪ٞ‬هغ ؽ‪ٞ‬اُ‪ ٢‬ثغ ِٓ‪ ٕٞ٤‬زائ ش‪٣ ٜ‬ب! ‪ ٞٛٝ‬وًج ‪ٝ‬وًض ٓ‪ٞ‬هغ رزْ ز‪٣‬ب ر‪ ٚ‬ك‪ ٢‬اُؾ ّ اُغبٓؼ‪٢‬‬ ‫ُغبٓؼخ ٗ‪ٌٓٞ٤‬ض‪ ٌٞ٤‬ؿبُ‪ٞ‬ة ‪ .‬كعال ػٖ ‪ٝ‬ع‪ َ٤ُ ٞ‬أٌُزجخ اُ هٔ‪٤‬خ ُِؼِ‪ ّٞ‬ك‪ ٢‬اُ اثػ اُزبُ‪:٢‬‬ ‫(‪،)http://fs.gallup.unm.edu//eBooks-otherformats.htm‬‬ ‫ٓغ اُؼم‪٣‬م ٖٓ اٌُزت ‪ٝ‬أُغالد اُؼِٔ‪٤‬خ أُ٘ر‪ ٞ‬ح اُز‪ ٢‬رظ‪ ٜ‬لثماػبر‪ ٚ‬اُؼِٔ‪٤‬خ‪ُٜٝ ،‬ب ؽ‪ٞ‬اُ‪1222 ٢‬‬ ‫ز‪٣‬ب ح ‪٤ٓٞ٣‬ب! ‪.‬‬ ‫‪ِٔ٣ٝ‬ي ٌٓزجخ هٔ‪٤‬خ ُِل٘‪ٝ ٕٞ‬ا‪ ٥‬اة اذ رعْ اُؼم‪٣‬م ٖٓ ًزج‪ ٝ، ٚ‬وُج‪ٓٞ‬بر‪ ٚ‬األ ث‪٤‬خ ‪ٝ‬اُل٘‪٤‬خ‬ ‫االثماػ‪٤‬خ‪ُٜٝ ،‬اا أُ‪ٞ‬هغ ٗؾ‪ 122 ٞ‬ز‪٣‬ب ح ك‪ ٢‬اُ‪ .ّٞ٤‬وٗظ اُ اثػ‬ ‫(‪)http://fs.gallup.unm.edu//eBooksLiterature.htm‬‬ ‫وصجؼ اُض‪ ٤‬كِ‪ٗ ٞ‬زٖ ذ‪ ٝ‬شؼج‪٤‬خ ًج‪ ٤‬ح ك‪ ٢‬عٔ‪٤‬غ وٗؾبء اُؼبُْ لذ وٕ وًض ٖٓ ‪ 1,222,222‬شخا‬ ‫ص٘‪٣ٞ‬ب ٖٓ ؽ‪ٞ‬اُ‪ 112 ٢‬ثِما ‪٣‬و‪ ٕٞٓٞ‬ثو اءح ‪ٝ‬رؾٔ‪ً َ٤‬زج‪ ٚ‬اإلٌُز ‪٤ٗٝ‬خ؛ ‪ٝ‬ؽبزد ًزج‪ ٚ‬ا‪٥‬الف ٖٓ‬ ‫اُز‪٣‬ب اد ش‪٣ ٜ‬ب‪.‬‬

‫‪01‬‬

‫‪Neutrosophic Science International Association‬‬

‫انفصم االول‬ ‫انرًهُذ‬ ‫(انًفهىو انُ​ُىذروصىفكٍ فٍ حضاب انرفاضم وانركايم)‬ ‫‪َ - 1.1‬ظرج ػايح‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ي ‪٣‬ؼ٘‪ ٢‬اصخ االكٌب ‪ٝ‬أُلب‪ ْ٤ٛ‬اُز‪ ٢‬ال رٌ‪ ٕٞ‬صؾ‪٤‬ؾخ ‪ٝ‬ال خبغئخ ‪ ٌُٖ ،‬ث‪ ٖ٤‬ذُي‪،‬‬ ‫‪ٛٝ‬اا ‪٣‬ؼ٘‪( ٢‬اُؾ‪٤‬ب ‪ ،‬اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ‪ ،‬اُال‪ٝ‬ظ‪ٞ‬ػ ‪ ،‬اُـٔ‪ٞ‬ض ‪ ،‬أُج‪ ، ْٜ‬اُالرٔبّ ‪ ،‬اُز٘بهط‪،‬‬ ‫‪ٝ‬ؿ‪ٛ ٤‬ب)‪ٝ ،‬لٕ ًَ ؽوَ ٖٓ ؽو‪ ٍٞ‬أُؼ كخ رِٔي عزئ‪ٜ‬ب اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬ذُي اُغزء اُا‪٣ ١‬ؾ‪١ٞ‬‬ ‫اُالرؼ‪ ،ٖ٤٤‬لذ رٔذ ‪ٝ‬ال ح أُ٘طن اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٝ ،٢‬أُغٔ‪ٞ‬ػخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪ ،‬واالؽزٔبُ‪٤‬خ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٝ ،‬االؽصبء اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٝ ، ٢‬اُو‪٤‬بس اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٝ ، ٢‬اُؾضبة اُزٔ‪٤ٜ‬م‪١‬‬ ‫ُِزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٝ ، ٢‬ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪...٢‬اُخ‪.‬‬ ‫كعال ػٖ ‪ٝ‬ع‪ ٞ‬اٗ‪ٞ‬اع ػم‪٣‬مح ٖٓ اُالرؼ‪ٛٝ – ٖ٤٤‬اا ٓب ‪٣‬ج‪ُ٘ ٖ٤‬ب‪ُٔ :‬بذا اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ي ‪ ٌٚ٘ٔ٣‬إ‬ ‫‪٣‬زط‪ ٞ‬ثط م ٓخزِلخ ؟‪.‬‬

‫‪ - 1.1‬انًمذيح‬ ‫اُغزء اال‪ٛ ٖٓ ٍٝ‬اا اٌُزبة ‪ً ٣‬ز ػِ‪ ٠‬روم‪ٓ ْ٣‬جب ئ ػِْ اُزلبظَ ‪ٝ‬اُزٌبَٓ‬ ‫‪ ٢ٛ‬اُخ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬الص‪ٔ٤‬ب اصخ اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ‪ ،‬كٔضال اُماُخ‬ ‫ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ رِٔي ثؼط اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ٓغ األخا ث٘ظ االػزجب رؼ ‪٣‬ق ٓ٘طِو‪ٜ‬ب لُ‪٠‬‬ ‫ٓما‪ٛ‬ب ‪ ،‬و‪ ٝ‬لُ‪ ٠‬اُؼالهخ اُز‪ ٢‬ررز ى ك‪ٜ٤‬ب ػ٘بص ٖٓ ‪ٓ A‬غ ػ٘بص ٖٓ ‪ً .B‬ؾبالد خبصخ‪ ،‬كوم‬ ‫هٔ٘ب ثزوم‪ ْ٣‬اُماُخ االص‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٝ‬اُماُخ اُِ‪ٞ‬ؿب ‪٣‬زٔ‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٝ ،‬رٌ‪ ٕٞ‬اُماُخ‬ ‫أُؼٌ‪ٞ‬س اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُز‪ٓ ٢ٛ ٢‬ؼٌ‪ٞ‬صب ُِماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪ٝ .‬ثبُٔضَ كبٕ اُ٘ٔ‪ٞ‬ذط‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٞٔٗ ٞٛ ٢‬ذط ُ‪ ٚ‬ثؼط اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) (أُج‪، ْٜ‬اُالر ً‪٤‬م‪ ،‬اُـٔ‪ٞ‬ض ‪ ،‬اُالرٔبّ‬ ‫‪ ،‬اُز٘بهط‪ٝ ،‬ؿ‪ٛ ٤‬ب)‪.‬‬ ‫اُغزء اُضبٗ‪ٛ ٖٓ ٢‬اا اٌُزبة ‪ً ٣‬ز ػِ‪ ٠‬ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٝ ،٢‬اُز‪ ٢‬رْ ٖٓ‬ ‫خالُ‪ٜ‬ب اصخ اُـب‪٣‬بد اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٝ‬االشزوبهبد ‪ٝ‬اُزٌبٓالد اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪.‬‬ ‫‪00‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ٗؾٖ ٗومّ ‪ٝ‬أل‪ ٓ ٍٝ‬ح ٓلب‪ ْ٤ٛ‬شج‪ ٚ‬اُـب‪٣‬خ ‪ٝ ،‬شج‪ ٚ‬االصزٔ ا ‪٣‬خ ‪ٝ،‬شج‪ ٚ‬أُرزوخ ‪ٝ ،‬شج‪ ٚ‬اُزٌبَٓ(‪ )1‬لُ‪٠‬‬ ‫عبٗت اُزؼب ‪٣‬ق اُزوِ‪٤‬م‪٣‬خ ُِـب‪٣‬خ ‪ٝ ،‬االصزٔ ا ‪٣‬خ ‪ٝ ،‬االشزوبم ‪ٝ‬اُزٌبَٓ رجبػب‪ُ ٌٖٔ٣ٝ .‬ؼِْ ٓجب ئ‬ ‫اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٝ،٢‬ػِْ ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ُٜٔ ٌٖٔ٣ ٢‬ب إ‬ ‫‪٣‬زط‪ ٞ‬إ ثط م ػم‪٣‬مح ‪٣ٝ ،‬ؼزٔم اخز‪٤‬ب غ ‪٣‬وخ اُزؼبَٓ ٓغ ٓرٌِخ ٓب ذاد ال رؾم‪٣‬م ػِ‪ ٠‬اٗ‪ٞ‬اع‬ ‫اُالرؾم‪٣‬م اُا‪ ١‬رٌِٔ‪ٓ ًَ ٚ‬رٌِخ ؽضت رطج‪٤‬وبر‪ٜ‬ب ‪.‬‬ ‫ٗومّ ك‪ٛ ٢‬اا اٌُزبة‪ ،‬ثؼط االٓضِخ ِ (اُالرؼ‪٘٤٤‬بد و‪ ٝ‬اُالرؾم‪٣‬ماد ) ‪ٝ‬غ م ٓزؼم ح ُِزؼبَٓ ٓغ ‪ٛ‬ا‪ٙ‬‬ ‫اُالرؼ‪٘٤٤‬بد اُخبصخ ‪٘ٛ ٌُٖ ،‬بى اُؼم‪٣‬م ٖٓ اُالرؾم‪٣‬ماد االخ ‪ ٟ‬أُ‪ٞ‬ع‪ ٞ‬ح ك‪ ٢‬ؽ‪٤‬بر٘ب اُ‪٤ٓٞ٤‬خ‬ ‫اصز‪ٜ‬ب ‪ٝ‬اػب ح ؽِ‪ٜ‬ب ثبصزخماّ غ م ٓربث‪ٜ‬خ ٗلض‪ٜ‬ب ا‪ ٝ‬ثبصزخماّ غ م اخ ‪ٟ‬‬ ‫‪ٝ‬اُز‪٣ ٢‬غت‬ ‫ٓخزِلخ اخ ‪ٓ ٟ‬خزِلخ‪ُٝ ،‬اُي كبٗ‪٣ ٚ‬غت اع اء اصبد ثؾض‪٤‬خ ٓزومٓخ اًض ك‪ ٢‬اُؾوَ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪. ٢‬‬

‫‪ - 3.1‬انفروق تٍُ ذحهُم انفررج ‪ ،‬وذحهُم انًجًىػح ‪ ،‬وانرحهُم انُ​ُىذروصىفكٍ‬ ‫يالحظح‬ ‫ك‪ ٢‬اُؾبُخ اُز‪ ٢‬ال ٗؼِْ‬ ‫ص‪٤‬زْ ك‪ٛ ٢‬اا اٌُزبة اػزٔب ص٘بخا ك‪ ٢‬االػزجب اُلز ح‬ ‫ك‪ٜ٤‬ب ً‪ ٕٞ‬ا‪٣‬ب ٖٓ ‪ a‬ا‪ ٞٛ b ٝ‬االًج ‪ ،‬ا‪ ٝ‬رِي اُؾبُخ اُز‪ ٢‬رِٔي ك‪ٜ٤‬ب اُلز ح ؿب‪٣‬بد ٓزلب‪ٝ‬رخ ٖٓ‬ ‫رٌ‪ٕٞ‬‬ ‫‪ ،‬ؽ‪٤‬ش ٗغم اٗ‪ُ ٚ‬جؼط اُو‪ ْ٤‬االً‪٤‬مح ٖٓ‬ ‫اُ‪ ٖٓٝ ٖ٤ٔ٤‬اُ‪٤‬ضب ثبُص‪٤‬ـخ‬ ‫‪.‬‬ ‫‪ُٝ‬و‪ ْ٤‬اخ ‪ ( ( ِ ٟ‬رٌ‪ٕٞ‬‬ ‫ذحهُم انفررج‬ ‫ك‪ ٢‬رؾِ‪ َ٤‬اُلز اد (ا‪ ٝ‬ؽضبة اُلز اد) رٌ‪ ٕٞ‬اُؼ٘بص ػجب ح ػٖ كز اد ثمال ٖٓ األ هبّ اُزوِ‪٤‬م‪٣‬خ‬ ‫أُزؼب ف ػِ‪ٜ٤‬ب ‪ٝ ،‬أُوص‪ ٖٓ ٞ‬اصخ رؾِ‪ َ٤‬اُلز ح ‪ ٞٛ‬رو ‪٣‬ت ثبُز‪٣‬ب ح ‪ٝ‬اُ٘وصبٕ ُالخطبء‬ ‫‪ٝ‬اُ٘برغخ ػٖ اُؼِٔ‪٤‬بد اُؾضبث‪٤‬خ ‪ٝ‬ػِ‪ ٠‬ذُي كبٕ اُخط ‪٣‬زْ رؾم‪٣‬م‪( ٙ‬اؽبغز‪ )ٚ‬ثبُلز ح أُـِوخ‪.‬‬ ‫ذحهُم انًجًىػح‬ ‫ك‪ ٢‬اُزؼ ‪٣‬ق اُضبثن ُزؾِ‪ َ٤‬اُلز ح لذا هٔ٘ب ثبصزجماٍ اُلز اد أُـِوخ ثٔغٔ‪ٞ‬ػبد اُز‪ً ٢‬بٗذ ثرٌَ‬ ‫كز اد ص٘ؾصَ ػِ‪ ٠‬رؼ ‪٣‬ق رؾِ‪ َ٤‬أُغٔ‪ٞ‬ػخ (ا‪ ٝ‬ؽضبة أُغٔ‪ٞ‬ػخ) ‪.‬‬ ‫ػِ‪ ٠‬صج‪ َ٤‬أُضبٍ ‪ُ ،‬زٌٖ ‪ h‬اُخ ٓغبُ‪ٜ‬ب ‪ٓٝ‬غبُ‪ٜ‬ب أُوبثَ ػجب ح ػٖ ٓغٔ‪ٞ‬ػبد ا‪ ٝ‬كز اد ا‪ ٝ‬اػما‬ ‫روِ‪٤‬م‪٣‬خ ‪ًٔٝ‬ب ‪٢ِ٣‬‬

‫‪From the Greek μερος, ‘part’. It is also used to define the theory of the relations of part‬‬ ‫‪to whole and the relations of part to part within a whole (mereology), started by‬‬ ‫‪Leśniewski, in “Foundations of the General Theory of Sets” (1916) and “Foundations of‬‬ ‫‪Mathematics” (1927–1931), continued by Leonard and Goodman's “The Calculus of‬‬ ‫‪Individuals” (1940),‬‬ ‫( ‪(1‬‬

‫‪02‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ؽ‪٤‬ش إ‬

‫‪ٓ ٢ٛ‬غٔ‪ٞ‬ػخ ًَ االػما اُؾو‪٤‬و‪٤‬خ ‪ٝ ،‬‬

‫‪ٓ ٢ٛ‬غٔ‪ٞ‬ػخ اُو‪ِ ٟٞ‬‬

‫‪.‬‬

‫= )‪h({1, 2, 3}) = {7, 9}, h([0, 1]) = (6, 8), h(-3‬‬ ‫‪{-1, -2}  (2.5, 8], h([x, x2]  [-x2, x]) = 0.‬‬

‫)‪(2‬‬

‫‪ٝ‬ػِ‪ ٠‬ذُي كبٕ رؾِ‪ َ٤‬أُغٔ‪ٞ‬ػخ ‪٣‬ؼم رؼٔ‪ٔ٤‬ب ُزؾِ‪ َ٤‬اُلز ح‪.‬‬ ‫انرحهُم انُ​ُىذروصىفكٍ (او انحضاب انُ​ُىذروصىفكٍ )‬ ‫‪ ٌٖٔ٣‬اػزجب اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬رؼٔ‪ٔ٤‬ب ٌُال اُزؾِ‪ ٖ٤ِ٤‬اُضبثو‪ ( ٖ٤‬رؾِ‪ َ٤‬اُلز ح‬ ‫‪ٝ‬رؾِ‪ َ٤‬أُغٔ‪ٞ‬ػخ)‪ ،‬ؽ‪٤‬ش إ اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٣ ٢‬زؼبَٓ ٓغ ًَ اٗ‪ٞ‬اع أُغٔ‪ٞ‬ػبد (ُ‪٤‬ش‬ ‫كوػ اُلز اد) ‪ ،‬كعال ػٖ رِي اُؾبُخ ػ٘مٓب ‪٘ٛ ٌٕٞ٣‬بى ثؼط اُالرؼ‪( ٖ٤٤‬ػِٔب إ اُالرؼ‪ٖ٤٤‬‬ ‫(اُالرؾم‪٣‬م)) هم ‪ ٌٕٞ٣‬ك‪ ٢‬أُغٔ‪ٞ‬ػبد ا‪ ٝ‬اُم‪ٝ‬اٍ ا‪ ٝ‬ك‪ٓ ٢‬لب‪ ْ٤ٛ‬اخ ‪ٓ ٟ‬ؼ كخ ػِ‪ٛ ٠‬ا‪ٙ‬‬ ‫أُغٔ‪ٞ‬ػبد)‪.‬‬ ‫ػ٘مٓب ٗزؼبَٓ ٓغ اُؼ٘بص ًٔغبٓ‪٤‬غ ثم‪ٝ ٕٝ‬ع‪ ٞ‬اُالرؾم‪٣‬م ‪ ،‬ػ٘م‪ٛ‬ب ص‪ ٌٕٞ٤‬اُزؾِ‪َ٤‬‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٓ ٢‬طبثوب ُزؾِ‪ َ٤‬أُغٔ‪ٞ‬ػخ‪ ُٞٝ ،‬رْ اُزؼبَٓ ٓغ اُؼ٘بص ثرٌَ كز اد كوػ ثمال ٖٓ‬ ‫أُغٔ‪ٞ‬ػبد ‪٘ٛ ٌٖ٣ ُْٝ‬بى ال رؼ‪ ٖ٤٤‬ػ٘م‪ٛ‬ب ص‪ ٌٕٞ٤‬اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٓ ٢‬طبثن ُزؾِ‪ َ٤‬اُلز ح‪،‬‬ ‫‪ٛ‬اا ‪ٝ‬ل ٕ ًبٕ ‪٘ٛ‬بُي ثؼط اُالرؼ‪ ، ٖ٤٤‬ػ٘م اصزخماّ اُلز اد ا‪ ٝ‬ػ٘م اصزخماّ أُغٔ‪ٞ‬ػبد ص‪٤‬زؾ‪ٍٞ‬‬ ‫اُزؾِ‪ َ٤‬ػ٘مئا اُ‪ ٠‬رؾِ‪ٞ٤ٗ َ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫ايصهح حىل انرحهُم انُ​ُىذروصىفكٍ‬ ‫ٓجمو ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٝ ٢‬ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫‪٣‬خزِلبٕ ػٖ رؾِ‪ َ٤‬أُغٔ‪ٞ‬ػخ ‪ ،‬الٗ‪ٜٔ‬ب ‪٣‬ضزخمٓبٕ اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ‪ًٔ .‬ضبٍ‪ ،‬ػ‪ٗٞ‬ب ٗ٘ظ‬ ‫اُم‪ٝ‬اٍ اُزبُ‪٤‬خ‪:‬‬ ‫‪f1(0 or 1) = 7.‬‬

‫)‪(3‬‬

‫‪٘ٛ‬بُي الرؼ‪ ٖ٤٤‬خبص ثٔغبٍ اُماُخ و‪ ١‬اٗ٘ب ؿ‪ٓ ٤‬ز ًم‪ ٖ٣‬ك‪ٔ٤‬ب اذا ًبٗذ ‪ًٔٝ‬ضبٍ اخ اُماُخ‬ ‫‪f1(0) = 7 or f1(1) = 7‬‬ ‫‪f2(2) = 5 or 6‬‬ ‫‪٘ٛ‬بُي الرؼ‪ ٖ٤٤‬خبص ثبُٔغبٍ أُوبثَ ُِماُخ ‪ٌٛٝ‬اا ٗؾٖ ُض٘ب ٓز ًم‪َٛ ٖ٣‬‬ ‫‪f2(2) = 5 or f2(2) = 6.‬‬

‫)‪(4‬‬ ‫‪ٝ‬ثص‪ ٞ‬ح اًض رؼو‪٤‬ما ‪.f3(-2 or -1) = -5 or 9‬‬ ‫‪٘ٛ‬بُي الرؼ‪ ٖ٤٤‬خبص ثٌال أُغبٍ ‪ٝ‬أُغبٍ أُوبثَ ُِماُخ ‪ .‬و‪ ١‬إ‬ ‫)‪(5‬‬

‫‪f3(-2) = -5, or f3(-2) = 9, or f3(-1) = -5, or f3(-1) = 9.‬‬ ‫‪03‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ٝ‬ثص‪ ٞ‬ح ػبٓخ ك ٕ‬ ‫‪fm,n(a1 or a2 or … or am) = b1 or b2 or … or bn.‬‬

‫)‪(6‬‬

‫لٕ اُم‪ٝ‬اٍ وػال‪ٝ ٙ‬اُز‪ ٢‬آزٌِذ ‪ٛ‬اا اُ٘‪ٞ‬ع ٖٓ اُالرؾم‪٣‬م رخزِق ػٖ اُم‪ٝ‬اٍ اُز‪ ٢‬و‪٘٣‬ب‪ٛ‬ب ك‪ٓ ٢‬ؼب ُخ‬ ‫(‪ٝ ) 1( ٝ )1‬اُز‪ً ٢‬بٕ ٓغبُ‪ٜ‬ب ‪ٓٝ‬غبُ‪ٜ‬ب أُوبثَ ػجب ح ػٖ ٓغبٓ‪٤‬غ ا‪ ٝ‬كز اد اً‪٤‬مح خبُ‪٤‬خ ٖٓ‬ ‫اُالرؾم‪٣‬م‪.‬‬ ‫ايصهح فٍ ذحهُم انًجًىػح‬ ‫اُماُخ ‪R‬‬

‫‪ f1: R‬رخزِق ك‪ٓ ٢‬غبُ‪ٜ‬ب ػٖ اُخ أُغٔ‪ٞ‬ػخ اُزبُ‪٤‬خ‪-:‬‬ ‫‪R, where g1({0, 1}) = 7.‬‬

‫)‪(7‬‬ ‫ًاا ٗغم إ اُماُخ ‪R‬‬

‫ا‪٣‬عب ٗ ‪R ٟ‬‬

‫‪ f2: R‬رخزِق ك‪ٓ ٢‬غبُ‪ٜ‬ب أُوبثَ‬ ‫‪R2, where g2(2) = {5, 6}.‬‬

‫)‪(8‬‬

‫‪g1: R2‬‬

‫‪g2: R‬‬

‫‪ f3: R‬رخزِق ك‪ٓ ٢‬غبُ‪ٜ‬ب ‪ٓٝ‬غبُ‪ ٚ‬أُوبثَ ؽ‪٤‬ش‬

‫)‪(9‬‬

‫‪R2, where g3({-2, -1}) = {-5, 9}.‬‬

‫‪g3: R2‬‬

‫‪ fm,n: R‬رخزِق ك‪ٓ ٢‬غبُ‪ٜ‬ب ‪ٓٝ‬غبُ‪ ٚ‬أُوبثَ ؽ‪٤‬ش ػٖ اُماُخ‬

‫‪ًٝ‬ؾبُخ ػبٓخ كبٕ اُماُخ ‪R‬‬ ‫‪gm,n : Rm Rn‬‬ ‫‪gm,n({a1, a2, …,am}) = {b1, b2, …, bn}.‬‬ ‫)‪(10‬‬ ‫‪ ٖٓٝ‬اُؾوبئن أُؼ ‪ٝ‬كخ إ و‪ٓ ١‬غٔ‪ٞ‬ػخ ‪ ٌٖٔ٣‬وٕ رٌ‪ٓ ٕٞ‬ؾز‪ٞ‬اح ك‪ ٢‬كز ح ٓـِوخ ‪ٓٝ ،‬غ ذُي كبٕ‬ ‫اُزؼبَٓ ٓغ اُلز اد اال‪ٝ‬صغ ‪ ٌٕٞ٣‬اصؼت ٖٓ اُزؼبَٓ ٓغ أُغبٓ‪٤‬غ اُع‪٤‬وخ ؽ‪٤‬ش ‪٣‬ؼط‪ٗ ٠‬زبئظ‬ ‫ػبٓخ ‪ ،‬ثَ االًض ٖٓ ذُي كبٗ‪ٜ‬ب رٌ‪ ٕٞ‬ؿ‪ ٤‬ه‪٤‬وخ‪ٝ .‬اُط ‪٣‬وخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُز‪ ٢‬رضزخمّ‬ ‫ٓغٔ‪ٞ‬ػبد اصـ ٓؾز‪ٞ‬ا‪ ٙ‬اخَ كز اد ‪ ٢ٛ ،‬غ ‪٣‬وخ اًض هخ ٖٓ رؾِ‪ َ٤‬اُلز ح‪.‬‬ ‫ايصهح فٍ ذحهُم انفررج‬ ‫ٗضزط‪٤‬غ اُو‪ ٍٞ‬إ اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٣ ٢‬زؼبَٓ ٓغ أُغبٓ‪٤‬غ اُز‪ ٢‬رؾ‪ ١ٞ‬ال رؾم‪٣‬م‪ٓ ،‬ضبٍ ذُي ٗغم‬ ‫اُؼ٘ص )‪٘٣ x(t,i,f‬زٔ‪ ٢‬ثرٌَ عزئ‪ُِٔ ٢‬غٔ‪ٞ‬ػخ ‪ٝ S‬ا‪٣‬عب اُؼ٘ص ٗلض‪ ٚ‬ال ‪٘٣‬زٔ‪ ٢‬ثرٌَ عزئ‪٢‬‬ ‫ُِٔغٔ‪ٞ‬ػخ ‪ ٖٓٝ S‬ع‪ٜ‬خ اخ ‪ ٟ‬كبٕ ‪ٛ‬اا اُؼ٘ص ‪٣‬ؾَٔ ثؼط اُالرؾم‪٣‬م ؽ‪٤‬ش رٌ‪ ٕٞ‬عخ اٗزٔبء‬ ‫اُؼ٘ص ‪ x‬ك‪ ٢‬أُغٔ‪ٞ‬ػخ ‪ S‬ؿ‪ٓ ٤‬ؾم ‪ٗ ،‬وصم ثاُي اٗ٘ب ال ِٗٔي ا ٗ‪ ٠‬كٌ ح ػٔب اذا ًبٕ ػ٘ص‬ ‫ٓؼ‪ٓ ٖ٤‬ضَ )‪٘٣ y(0,1,0‬زٔ‪ ٢‬و‪ ٝ‬ال ‪٘٣‬زٔ‪ ٢‬اُ‪ ٠‬أُغٔ‪ٞ‬ػخ (رٔبّ اُالرؼ‪)ٖ٤٤‬؟ ‪ ،‬ا‪ ٝ‬هم رؼ٘‪ ٢‬اٗ‪ٞ٣ ٚ‬عم‬ ‫ػ٘ص ‪٘٣‬زٔ‪ ٢‬اُ‪ ٠‬أُغٔ‪ٞ‬ػخ ‪٘​ٌُ٘ٝ S‬ب ال ٗضزط‪٤‬غ رؾم‪٣‬م غج‪٤‬ؼز‪ٜ‬ب ‪.‬إ ًال ٖٓ رؾِ‪ َ٤‬اُلز ح ‪ٝ‬رؾِ‪َ٤‬‬ ‫أُغٔ‪ٞ‬ػخ ػبعز‪ ٖ٣‬ػٖ اُزؼبَٓ ٓغ ‪ٛ‬اا اُ٘‪ٞ‬ع ٖٓ االٗزٔبء‪.‬‬ ‫‪04‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ُ٘ل ض إ ُم‪٘٣‬ب اُلز ح اُزبُ‪٤‬خ [ )‪ L=[0, 5(0.6, 0.1, 0.3‬ؽ‪٤‬ش إ اُؼم )‪٘٣ 5(0.6, 0.1, 0.3‬زٔ‪ ٢‬عزئ‪٤‬ب‬ ‫ثم عخ (‪ُِ )0.6‬لز ح ‪ٝ ، L‬ك‪ ٢‬اُ‪ٞ‬هذ ٗلض‪ ٚ‬ال‪٘٣‬زٔ‪ ٢‬ثرٌَ عزئ‪ُِ ٢‬لز ح ‪ L‬ثم عخ (‪ًٔ ، )0.3‬ب‬ ‫‪ُِ ٌٕٞ٣‬ؼم ‪ 5‬عخ الٗؾم‪٣‬م ك‪ ٢‬االٗزٔبء اُ‪ L ٠‬ثو‪ٔ٤‬خ (‪.)0.1‬‬ ‫]‪ٝ . L ≠ [0, 5) ٝ L ≠ [0, 5‬ك‪ ٢‬اُؾو‪٤‬وخ كبٕ ‪ L‬روغ ث‪ ٖ٤‬اُلز ر‪.ٖ٤‬‬ ‫ا‪ ١‬إ ‪.‬‬ ‫]‪[0, 5) ⊂ L ⊂ [0, 5‬‬

‫)‪(11‬‬

‫ذُي الٕ اُؼ٘ص ‪ 5‬ال ‪٘٣‬زٔ‪ ٢‬اُ‪ ٠‬اُلز ح )‪ًٔ ، [0,5‬ب إ اٗزٔبئ‪ ٌٕٞ٣ ٚ‬عزئ‪٤‬ب ُِلز ح ‪[0, 5(0.6,‬‬ ‫[)‪ٝ 0.1, 0.3‬ثبُز ً‪٤‬م ‪٘٣‬زٔ‪ ٢‬اُ‪ ٠‬اُلز ح ]‪ٝ . [0,5‬ػِ‪ ٠‬ذُي رٌ‪ ٕٞ‬اُلز ح ‪ ٢ٛ L‬عزءا ٖٓ اُزؾِ‪َ٤‬‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤ُٝ ٢‬ش رؾِ‪ َ٤‬اُلز اد‪.‬‬ ‫ػز‪٣‬ز‪ ١‬اُوب ئ ‪ ،‬الؽع اُم‪ٝ‬اٍ اُزبُ‪٤‬خ‬ ‫‪k1( [0, 5] ) = [-4, 6], or k2( [-2, -4] ) = [0, 5],‬‬ ‫)‪(12‬‬ ‫اُم‪ٝ‬اٍ ‪٘٣ k2 ٝ k1‬زٔ‪٤‬بٕ ُزؾِ‪ َ٤‬اُلز ح ‪ .‬آب اُم‪ٝ‬اٍ‬ ‫‪k3([0, 5(0.6, 0.1, 0.3)[)=[-4, 6], or k4([-2, -4])=[0, 5(0.6,0.1,0.3)[,‬‬ ‫)‪(13‬‬ ‫ك‪ ٢ٜ‬ثم‪ ٕٝ‬شي ر٘زٔ‪ ٢‬اُ‪ ٠‬اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪، ٢‬‬ ‫‪ ٢ٛ‬اُمُخ اُز‪ ٢‬رِٔي ثؼط اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ٓغ األخا‬ ‫لٕ اُمُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ث٘ظ االػزجب رؼ ف ٓغبُ‪ٜ‬ب اُ‪ٓ ٠‬غبُ‪ٜ‬ب أُوبثَ ‪ ،‬ا‪ ٝ‬اُ‪ ٠‬ػالهز‪ٜ‬ب اُز‪ ٢‬رر ى ث‪ٜ‬ب ػ٘بص ٖٓ ‪A‬‬ ‫ٓغ ػ٘بص ٖٓ ‪ ،B‬ا‪ٓ ٝ‬غ االخا ث٘ظ االػزجب اص٘بٕ و‪ٝ‬صالصخ ٖٓ االػزجب اد اػال‪ٓ ٙ‬ؼب ‪.‬‬ ‫ك‪ ٢‬رؾِ‪ َ٤‬اُلز ح ‪٣‬زْ كوػ اصخ اُم‪ٝ‬اٍ أُؼ كخ ػِ‪ ٠‬كز اد ‪ٝ ،‬اُز‪ ٢‬ه‪ٜٔ٤‬ب كز اد ا‪٣‬عب ك‪ ٢‬ؽ‪ٖ٤‬‬ ‫اٗ‪ٜ‬ب خبُ‪٤‬خ ٖٓ اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ‪ُ .‬اُي ‪ٗ ،‬غم إ اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬اًض ػٔ‪٤ٓٞ‬خ ٖٓ‬ ‫رؾِ‪ َ٤‬اُلز ح ‪ .‬ا‪٣‬عب‪ ،‬إ اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٣ ٢‬زؼبَٓ ٓغ اُالرؼ‪ٓ ٖ٤٤‬غ األخا ث٘ظ االػزجب‬ ‫ٓغبٍ اُماُخ ‪ٓ ،‬غبُ‪ٜ‬ب أُوبثَ ‪ ،‬ا‪ً ٝ‬ال‪ٔٛ‬ب ‪ .‬خا ٓضال اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُزبُ‪٤‬خ ‪-:‬‬ ‫)‪(14‬‬ ‫‪.‬وذ إ أُ ًجخ ‪ I‬رٔضَ اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م)‬ ‫)‪(15‬‬ ‫)‪(16‬‬ ‫)‪(17‬‬ ‫)‪(18‬‬

‫}{‬

‫}{‬

‫;‬

‫ك‪ ٢‬اُماُخ ‪٣ k‬لرَ اخزجب أُضزو‪ ْ٤‬اُؼٔ‪ ١ ٞ‬اُزوِ‪٤‬م‪ُِ٘ٔ ١‬ؾ٘‪٤‬بد اُز‪ ٢‬رٔضَ ‪ٝ‬اٍ روِ‪٤‬م‪٣‬خ‪.‬‬ ‫)‪(19‬‬ ‫اذٕ ُم‪٘٣‬ب رؾِ‪ َ٤‬اُلز ح ⊃ رؾِ‪ َ٤‬أُغٔ‪ٞ‬ػخ ⊃ اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫‪05‬‬

Neutrosophic Science International Association

‫انرىذر انًرضاو‬ َ٤ِ‫اُزؾ‬ٝ ‫ػخ‬ٞٔ‫َ أُغ‬٤ِ‫ رؾ‬٠ِ‫ ػ‬ٚ‫و‬٤‫ٌٖٔ رطج‬٣ ‫َ اُلز ح‬٤ِ‫ ُزؾ‬١ٝ‫ر أُزضب‬ٞ‫الط اُز‬٣‫ّ ا‬ٜٞ‫إ ٓل‬ ‫ هضٔخ‬ٝ‫ ظ ة ا‬ٝ‫ غ ػ ا‬ٝ‫خ عٔغ ا‬٤ِٔ‫ٔضَ ػ‬٣ ʘ ‫ ًبٕ اُ ٓز‬ُٞ ، ‫ٌاا‬ٛٝ . ٢ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫ا‬ ٕ‫ كب‬A ⊆ C and B ⊆ D ٕ‫ش ا‬٤‫غ ثؾ‬٤ٓ‫ ا ثغ ٓغب‬A,B,C,D ‫ًبٗذ‬ٝ ‫ػبد‬ٞٔ‫أُغ‬ A ʘ B ⊆ C ʘ D.

(20) . ‫َ اُلز ح‬٤ِ‫ٓجبش ُزؾ‬ٝ ٢ُٝ‫بٕ ا‬ٛ ‫اُج‬ ٕ‫ش ا‬٤‫ ثؾ‬a ∈ A and b ∈ B ‫عم‬ٞ‫ ػ٘مئا ر‬x ∈ A ʘ B ٕ‫ ُ٘ل ض ا‬٢ِ٣ ‫كٌٔب‬ ‫٘ب‬٣‫خ ُم‬ٜ‫وخ ٓربث‬٣ ‫ ثط‬a ∈ C ٕ‫ كب‬a ∈ A and A ⊆ C ٕ‫ ثٔب ا‬. x = a ʘ b b ∈ D ٢٘‫ؼ‬٣ b ∈ B and B ⊆ D x = a ʘ b ∈ C ʘ D ٕ‫ كب‬٢ُ‫ثبُزب‬ٝ ‫بخا ث٘ظ‬٣ ٕ‫ اُوب ئ ا‬٠ِ‫غت ػ‬٣ ٌُٖ ٢ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫َ ا‬٤ِ‫ اُزؾ‬٢‫ؾب ك‬٤‫ٕ صؾ‬ٌٞ٣ ٙ‫بٕ اػال‬ٛ ‫إ اُج‬ ‫ أُ ًجبد‬٢ِ٣ ‫ًٔب‬ٝ ‫خ ٓضال‬٤ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫خ ا‬٤‫بد اُؾضبث‬٤ِٔ‫اؽمح ٖٓ اُؼ‬ٝ ٖ٤ٔ‫االػزجب رع‬ ٢‫ ك‬ٙ‫ا‬ٞ‫ ٓؾز‬M ‫خ‬٤ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫ػخ ا‬ٞٔ‫٘ب أُغ‬٣‫ ُم‬t,I,f )‫خ‬٣‫م‬٤ِ‫رخ (اُزو‬ُٜ‫خ ا‬٤‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫ا‬ ٢ِ٣ ‫ اذا رؾون ٓب‬N ‫خ‬٤ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫ػخ ا‬ٞٔ‫أُغ‬ ٕ‫ ٓغ ٓالؽظخ ا‬x(tn,in,fn) ∈ N ٌٕٞ‫ ر‬x(tM,iM,fM) ∈ M ‫ ػ٘ص‬١‫ال‬ tM ≤ tN, iM ≥ iN, and fM ≥ fN

‫االصرُراض‬ ّ‫خ ػب‬٤ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫خ ا‬٤ُ‫ االؽزٔب‬٢‫ اٗغزد ك‬٢‫س اُز‬ٞ‫ذط رِي اُجؾ‬ُٞٔ٘ ٚ‫اا اُجؾش ٓربث‬ٛ ٕ‫ا‬ . ٙ‫ ا ٗب‬٢‫ب ك‬ٜ‫ صزغم ثؼع‬٢‫) ٖٓ أُصب اُز‬1214( ‫ ص٘خ‬٢ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫االؽصبء ا‬ٝ )1211(

‫انًصادر‬ 1. Florentin Smarandache, Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability, Sitech & Educational, Craiova, Columbus, 140 p., 2013.

2. Florentin Smarandache, Introduction to Neutrosophic Statistics, Sitech and Education Publisher, Craiova, 123 p., 2014.

3. Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, Introduction to Interval Analysis, Society of Industrial and Applied Mathematics, Philadelphia, PA, USA, 2009.

4. Dilwyn Edwards and Mike Hamson, Guide to Mathematical Modelling, CRC Press, Boca Raton, 1990.

06

‫‪Neutrosophic Science International Association‬‬

‫‪ – 4.1‬انًماَُش انهُذصُح االونُح غُر انًؼُ​ُح‬ ‫إ اُ ‪٣‬بظ‪٤‬بد ثزؼج‪ ٤‬ؿ‪ٓ ٤‬ؼ‪٣ ٖ٤‬ضٔ‪ ٠‬اُؾضبة اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ .٢‬أُؼ٘‪ ٠‬اُؾو‪٤‬و‪ ِ ٢‬اُالرؼ‪ٖ٤٤‬‬ ‫(اُالرؾم‪٣‬م) ‪ ٞٛ‬ػمّ اُمهخ ‪ ،‬ػمّ اُ‪ٞ‬ظ‪ٞ‬ػ ‪ ،‬أُج‪ ، ْٜ‬اُالرٔبّ ؿ‪ٓ ٤‬ز٘بصن (ٓز٘بك ) ‪ٓ ،‬ؼِ‪ٓٞ‬بد‬ ‫ٓز٘بهعخ‪ .‬ث‪ٔ٘٤‬ب اُؾضبة اُزوِ‪٤‬م‪٣ ١‬صق ‪٘٣‬بٓ‪٤ٌ٤‬خ ػبُٔ٘ب ‪ٗ ،‬غم وٕ اُؾضبة اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٣ ٢‬صق‬ ‫اُالرؼ‪( ٖ٤٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ي) ك‪ٛ ٢‬ا‪ ٙ‬اُم‪٘٣‬بٓ‪٤ٌ٤‬خ‪ .‬اُؾضجبٕ اُزوِ‪٤‬م‪٣ ١‬زؼبَٓ ٓغ ٓلب‪ٓ ْ٤ٛ‬ضَ (أُ‪، َ٤‬‬ ‫خػ أُ​ٔبس‪ ،‬غ‪ ٍٞ‬اُو‪ٞ‬س‪(،‬أُ ًز) ٗوطخ اُزٔ ًز ‪ ،‬عخ االٗؾ٘بء ‪ٝ‬اُزو‪ٞ‬س ‪ ،‬أُضبؽخ ‪،‬‬ ‫اُؾغْ‪ً ،‬اا اُض ػخ ‪ٝ‬اُزؼغ‪ًٔ )َ٤‬وب‪٤٣‬ش ٓعج‪ٞ‬غخ‪.‬‬ ‫ػِٔب اٗ‪ ٚ‬ك‪ ٢‬ؽ‪٤‬بر٘ب اُ‪٤ٓٞ٤‬خ ‪٘ٛ‬بُي ؽبالد ػم‪٣‬مح ٗزؼبَٓ ك‪ٜ٤‬ب ٓغ ٓوب‪٤٣‬ش رو ‪٣‬ج‪٤‬خ ‪.‬اُؾضجبٕ اُزٔ‪٤ٜ‬م‪١‬‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ ‪ ٢‬اًض صجبرب ‪٣ ٞٛٝ‬زؾمس ػٖ اُـٔ‪ٞ‬ض اُضبًٖ ا‪ ٝ‬االٓ‪ ٞ‬أُج‪ٜٔ‬خ اُضبثزخ‪ .‬ك‪٢‬‬ ‫اُؾضجبٕ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٣ ٢‬زْ اُزؼبَٓ ٓغ االكٌب اُز‪ ٢‬رِٔي ال رؼ‪ٝ ٖ٤٤‬اًض ٖٓ ذُي ‪ ،‬اُالرؼ‪ٖ٤٤‬‬ ‫(اُالرؾم‪٣‬م) ‪ٝ‬ػمّ اُز‪ٞ‬ك‪٤‬ن ‪ٝ ،‬اُزؼٔ‪ٔ٤‬بد اُز‪ٗ ٌٖٔ٣ ٢‬وِ‪ٜ‬ب ٖٓ ػِٔ‪٤‬خ اُ‪ ٠‬اخ ‪.ٟ‬‬ ‫ًِٔخا ُٔب ذً اػال‪ ٙ‬ك‪ ٢‬اُؼبُْ أُضبُ‪( ٢‬أُغزٔغ أُضبُ‪ ) ٢‬ر‪ٞ‬عم ‪ٝ‬اكغ (ٓوبصم) ‪ٝ‬اكٌب ٓضبُ‪٤‬خ‬ ‫اُز‪ ٢‬ك‪ٜ٤‬ب ‪٣‬زْ اصزخماّ ؽضجبٕ اُزلبظَ ‪ٝ‬اُزٌبَٓ اُزوِ‪٤‬م‪.١‬‬ ‫)‬ ‫ٓضال‪ٗ ،‬غم إ اُزو‪ٞ‬س ك‪ ٢‬اُمائ ح أُضبُ‪٤‬خ ذاد ٗصق اُوط ‪ ٞٛ r > 0‬ػم صبثذ (‪٣‬ضب‪١ٝ‬‬ ‫ث‪ٔ٘٤‬ب ثبُ٘ضجخ ُِمائ ح ؿ‪ ٤‬أُضبُ‪٤‬خ كبٕ اٗؾ٘بئ‪ٜ‬ب ‪ ٌٖٔ٣‬رٔض‪ ِٚ٤‬ثلز ح ٓؾز‪ٞ‬ا‪ ٙ‬ك‪٢‬‬ ‫‪ ٞٛٝ ε‬هْ صـ‪ ( ٤‬ه‪٤‬ن)‪.‬‬ ‫ؽ‪٤‬ش‬ ‫‪ٝ ،‬اُا‪ٔ٣ ١‬ضَ ع‪ٞ‬ا اُؼم‬ ‫‪ε‬‬ ‫‪ε‬‬ ‫ػِ‪٠‬‬ ‫ك‪ ٢‬أُضِش أُضبُ‪ ٢‬هبئْ اُزا‪٣ٝ‬خ ذ‪ ٝ‬االظالع ‪ ٌٕٞ٣ 1 cm , 2 cm‬اُ‪ٞ‬ر ٓضب‪٣ٝ‬ب ٍ ‪cm‬‬ ‫الٕ‬ ‫و‪ ١‬ؽبٍ ٗغم اٗ‪ ٚ‬ك‪ ٢‬ػبُٔ٘ب ؿ‪ ٤‬أُضبُ‪ ٢‬ال ٗضزط‪٤‬غ صْ هطؼخ أُضزو‪٣ ْ٤‬ضب‪ ١ٝ‬ثبُعجػ‬ ‫‪ ٞٛ‬ػم ؿ‪ٗ ٤‬ضج‪ِٔ٣ ٢‬ي ػم ال ٗ‪ٜ‬بئ‪ ٖٓ ٢‬اال هبّ اُؼر ‪٣‬خ ُاا ٗؾٖ ثؾبعخ اُ‪ ٠‬رو ‪٣‬ج‪ٜ‬ب‬ ‫اُؼم‬ ‫اُ‪ ٠‬ثؼط اال هبّ اُؼر ‪٣‬خ‬ ‫? ‪√5‬‬

‫شٌَ (‪)1‬‬ ‫) ‪ٔٛ‬ب‬ ‫(ػِٔب إ‬ ‫‪ٝ‬‬ ‫‪ ،‬إ‬ ‫إ ٓضبؽخ اُوطغ اُ٘بها أُضبُ‪ٞٛ ٢‬‬ ‫أُؾ‪ ٖ٣ ٞ‬االصبص‪ٝ ٢‬اُل ػ‪ُِ ٢‬وطغ اُ٘بها ػِ‪ ٠‬اُز‪ٞ‬اُ‪ٗ ، ٢‬الؽع اٗ٘ب ٗضزط‪٤‬غ رٔض‪ٛ َ٤‬اا اُوطغ‬ ‫ثمهخ الٕ ػم ٓزضبّ (‪ ( )transcendental number‬و‪ ١‬اٗ‪ ٚ‬ال ‪ٔ٣‬ضَ ؽال ُٔؼب الد ثٔزؼم ح‬ ‫ؽم‪ ٝ‬ذاد ٓؼبٓالد ٓؼو‪ُٞ‬خ ) ًٔب ‪ُٝ‬م‪٘٣‬ب ػم الٗ‪ٜ‬بئ‪ ٖٓ ٢‬اال هبّ اُؼر ‪٣‬خ ‪.‬‬ ‫اك ض‬

‫‪ٝ‬‬

‫صزٌ‪ٓ ٕٞ‬ضبؽخ اُوطغ ‪cm2‬‬ ‫‪07‬‬

‫‪Neutrosophic Science International Association‬‬

‫شٌَ (‪)1‬‬ ‫ٌُ٘​٘ب ال ٗضزط‪٤‬غ إ ٗعٖٔ ثبُعجػ أُضبؽخ اخَ ‪ٛ‬اا اُوطغ اُ٘بها الٕ‬ ‫هٔب ٓعج‪ٞ‬غب ‪ .‬ثاُي‪ ،‬ك٘ؾٖ ٗؼَٔ ثرٌَ رو ‪٣‬ج‪( ٢‬ؿ‪ ٤‬ه‪٤‬ن ‪ ،‬ؿ‪ٓ ٤‬ؼ‪.)ٖ٤‬‬ ‫‪ٝ‬ثرٌَ ٓربث‪ ، ٚ‬ثبُ٘ضجخ ُؾغْ اٌُ ح أُضبُ‪٢‬‬ ‫ػ٘مئا كبٕ‬

‫ال ‪٣‬ؼزج‬

‫لذ لٕ ٗصق اُوط ‪ r ٞٛ‬ػ٘مٓب‬ ‫‪ ٞٛٝ‬ػم ٓزضبّ ُ‪ ٚ‬ػم الٗ‪ٜ‬بئ‪ٖٓ ٢‬‬

‫أُ ارت اُؼر ‪٣‬خ ‪ٌٛٝ .‬اا ك٘ؾٖ ؿ‪ ٤‬هب ‪ ٖ٣‬ػِ‪ ٠‬اُؾص‪ ٍٞ‬ػِ‪ ٠‬ؽغْ اٌُ ح ك‪ ٢‬ا ٗب‪. ٙ‬‬

‫شٌَ (‪)1‬‬ ‫‪ٝ‬اُؾغْ ‪٣‬ضب‪١ٝ‬‬

‫‪08‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ – 5.1‬لىاٍَُ فُزَائُح غُر يؼُ​ُح‬ ‫إ اُزطج‪٤‬وبد اُل‪٤‬زائ‪٤‬خ ُِ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ي عِ‪٤‬خ ُِؼ‪٤‬بٕ ‪ ،‬لذ لٕ اُؼم‪٣‬م ٖٓ اُو‪ٞ‬اٗ‪ ٖ٤‬اُل‪٤‬ز‪٣‬بئ‪٤‬خ‬ ‫ٓؼ كخ ك‪ٗ ٢‬ظْ ٓـِوخ ؽب ح ( اُ٘ظْ أُضبُ‪٤‬خ)(‪ )1‬ػِٔب إ ‪ٛ‬ا‪ ٙ‬اُ٘ظْ ؿ‪ٞٓ ٤‬ع‪ ٞ‬ح ك‪ ٢‬ػبُٔ٘ب‪ ،‬ك‪٢‬‬ ‫اُؾو‪٤‬وخ ٗؾٖ ٗزؼبَٓ ٓغ اُ٘ظْ أُـِوخ رو ‪٣‬ج‪٤‬ب كوػ ٓٔب ‪٣‬ؤ ‪ ١‬اُ‪ ٠‬ؽم‪ٝ‬س كغ‪ٞ‬ح رٌٔ٘​٘ب ٖٓ اصزخماّ‬ ‫اُ٘ظ ‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ (ٗظ ‪٣‬خ اُالرؾم‪٣‬م)‪ُٜ .‬اا كبٕ اُ٘ظبّ هم ‪ٓ ٌٕٞ٣‬ـِن ثٔوما ‪ ( t%‬ك‪٢‬‬ ‫اؿِت االؽ‪٤‬بٕ رٌ‪ٗ ُٚ ) t<100 ٕٞ‬ضجخ ‪ ٖٓ i%‬اُالرؼ‪ ٖ٤٤‬ك‪ ٢‬اؿالم ‪ٛ‬اا اُ٘ظبّ ا‪ ٝ‬اٗلزبؽ‪ٚ‬‬ ‫(‪ًٔ )closeness or openness‬ب إ اُ٘ظبّ ‪ٓ ٌٕٞ٣‬لز‪ٞ‬ؽب ثٔوما ‪ f%‬ثبُ٘ز‪٤‬غخ ‪ ،‬كبٕ ه‪ٞ‬اٗ‪ٖ٤‬‬ ‫اُل‪٤‬ز‪٣‬بء اُ٘ظ ‪٣‬خ هم ‪ ٌٕٞ٣‬صؾ‪٤‬ؾب ك‪ ٢‬ؽ‪٤‬بر٘ب اُؼِٔ‪٤‬خ ثبهَ ٖٓ ‪ٌٛٝ 100%‬اا كبٕ اُوبٗ‪ٌٖٔ٣ ٕٞ‬‬ ‫إ ‪ِٔ٣‬ي ٗضجخ اخ ‪ ٖٓ ٟ‬اُالرؼ‪ًٔ( ٖ٤٤‬ب ك‪ ٢‬أُ٘طن اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪.) ٢‬‬ ‫ث‪ٝ ٖ٤‬ع‪ٝ ٞ‬ػمّ ‪ٝ‬ع‪ ٞ‬اُوبٗ‪ ٕٞ‬اُ٘ظ ‪( ١‬اُلٌ ح) ك‪ ٢‬اُزطج‪٤‬ن اُؼِٔ‪ ٌٖٔ٣ ، ٢‬رعٔ‪ٓ ٖ٤‬ز‪ٞ‬صطبد –‬ ‫ٓزؼم ح ‪ ،‬و‪ ١‬اُؾبالد اُز‪ ٌٕٞ٣ ٢‬ك‪ٜ٤‬ب اُوبٗ‪ ٕٞ‬اُ٘ظ ‪( ١‬اُلٌ ح) ٓ‪ٞ‬ع‪ ٞ‬ثرٌَ عزئ‪ٝ ٢‬ؿ‪ٞٓ ٤‬ع‪ٞ‬‬ ‫عزئ‪٤‬ب ا‪٣‬عب‪.‬‬

‫(‪(2‬‬

‫‪Fu Yuhua, “Pauli Exclusion Principle and the Law of Included Multiple-Middle”, in‬‬ ‫‪Neutrosophic Sets and Systems, Vol. 6, 2014.‬‬

‫‪09‬‬

‫‪Neutrosophic Science International Association‬‬

‫انفصم انصاٍَ‬ ‫يثادئ انرفاضم وانركايم انُ​ُىذروصىفكٍ‬ ‫‪ - 1.2‬انؼًهُاخ انجثرَح نهًجايُغ‬ ‫ُزٌٖ‬ ‫)‪(21‬‬ ‫)‪(22‬‬ ‫)‪(23‬‬ ‫)‪(24‬‬ ‫)‪(25‬‬

‫‪ٓ ٝ‬غٔ‪ٞ‬ػزبٕ‬

‫∈‬

‫‪ ٞٛ‬و‪ ١‬ه‪٤‬بص‪ ٢‬ػ٘مئا‬ ‫{‬

‫;} ∈ |‬ ‫{‬ ‫∈ |‬ ‫;} ∈‬ ‫{‬ ‫∈ |‬ ‫;} ∈‬ ‫∈ | {‬ ‫;} ∈‬ ‫‪-‬‬

‫∈‬

‫∈ | ‪,‬‬

‫‪ -2.2‬انؼاللاخ تٍُ انًجايُغ انجزئُح انُ​ُىذروصىفكُح‬ ‫ُزٌٖ ػالهخ ٓغٔ‪ٞ‬ػخ عزئ‪٤‬خ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ُٔغٔ‪ٞ‬ػز‪ٔٛ ٖ٤‬ب ‪ٗ ٝ‬ؼ ف ‪ r‬ػِ‪ ٠‬وٗ‪ٜ‬ب اُز‪ٝ‬ط‬ ‫ٓغٔ‪ٞ‬ػخ عزئ‪٤‬خ ٖٓ ٓغ‬ ‫ٓغٔ‪ٞ‬ػخ عزئ‪٤‬خ ٖٓ ‪ٝ‬‬ ‫لذ لٕ‬ ‫أُ رت ثبُص‪٤‬ـخ‬ ‫ثؼط اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ‪.‬‬ ‫‪ٝ‬اُا‪٘٣ ١‬زٔ‪٢‬‬ ‫لٕ اُؼالهخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثبإلظبكخ اُ‪ٝ ٠‬ع‪ٓ ٞ‬ؤًم ُِز‪ٝ‬ط أُ رت‬ ‫ٓغٔ‪ٞ‬ػخ‬ ‫ؽ‪٤‬ش‬ ‫ثٔب ‪٣‬ؾ‪ ١ٞ‬ا‪٣‬عب ز‪ٝ‬ط ٓ رت ٓؾزَٔ ‪ٞٛ‬‬ ‫ثم عخ ‪ 100%‬اُ‪٠‬‬ ‫ٓغٔ‪ٞ‬ػخ عزئ‪٤‬خ ٖٓ ‪ٝ ،‬اُز‪ ٢‬ثٔب ‪ ٌٖٔ٣‬إ ر٘زٔ‪ ٢‬اُ‪٘​ٌُ٘ r ٠‬ب ال ٗؼِْ عخ‬ ‫عزئ‪٤‬خ ٖٓ ‪ٝ‬‬ ‫ؽ‪٤‬ش‬ ‫اٗزٔبئ‪ٜ‬ب‪ ،‬ا‪ ٝ‬هم ر٘زٔ‪ٝ ٢‬ثرٌَ عزئ‪ ٢‬اُ‪ r ٠‬ثو‪ٔ٤‬خ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪٢ٛ‬‬ ‫رؼ٘‪ ٢‬عخ االُزؾبم ثع ‪ ٝ ، r‬رؼ٘‪ ٢‬عخ ؿ‪ٓ ٤‬ؼ‪٘٤‬خ ٖٓ االُزؾبم ث‪ٔ٘٤‬ب رؼ٘‪ ٢‬عخ ػمّ‬ ‫االُزؾبم ة ‪.‬‬ ‫يصال‬ ‫{‬

‫{ }‬ ‫}‬ ‫} {} { } {} {‬ ‫)‪(26‬‬ ‫{‬ ‫}‬ ‫} {} {‬ ‫} {} {‬ ‫ؽ‪٤‬ش } { } { ‪٘٣ { } { } ٝ‬زٔ‪٤‬بٕ ثرٌَ اً‪٤‬م اُ‪ r ٠‬ث‪ٔ٘٤‬ب } { } { اٗزٔبئ‪ٚ‬‬ ‫ٖٓ اُالرؼ‪( ٖ٤٤‬اُالرؼ‪ُِٔ )ٖ٤٤‬ؾو‪٤‬خ ‪ٛ‬اا اُؼ٘ص اُ‪ً ٠‬اُي‬ ‫‪ٝ‬‬ ‫ث٘ضجخ‬ ‫عزئ‪ ٢‬اُ‪٠‬‬ ‫ٖٓ ػمّ االٗزٔبء اُ‪ٝ ، ٠‬إ } { } { ‪ ٞٛ‬ز‪ٝ‬ط ٓ رت ٓؾزَٔ ( ثٔب ‪٘٣‬زٔ‪ ٢‬اُ‪٠‬‬ ‫‪٘​ٌُ٘ ، r‬ب ال ٗؼِْ ثب‪ ١‬عخ )‪.‬‬

‫‪21‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ -3.2‬دانح انًجًىػح انجزئُح انُ​ُىذروصىفكُح‬ ‫‪ ٢ٛ‬ػالهخ ٓغٔ‪ٞ‬ػخ عزئ‪٤‬خ‬ ‫اُخ أُغٔ‪ٞ‬ػخ اُغزئ‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪،‬‬ ‫ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثؾ‪٤‬ش اذا ًبٕ ‪٘ٛ‬بى ٓغٔ‪ٞ‬ػخ عزئ‪٤‬خ ⊆ ٓغ‬ ‫(إ ذُي ‪ٔ٣‬ضَ اخزجب أُضزو‪ ْ٤‬اُؼٔ‪ ١ ٞ‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ ٢‬اُا‪٣ ١‬زْ ر‪ٞ‬ص‪٤‬ؼ‪ٖٓ ٚ‬‬ ‫ػ٘مئا‬ ‫أُغبٍ أُوبثَ اُ‪ٜ‬ش اُا‪ ١‬ػ٘بص ‪ٓ ٙ‬غبٓ‪٤‬غ ) ‪.‬‬ ‫‪ًٝ‬ؾبُخ خبصخ ‪ ٌٖٔ٣‬اػزجب إ اُؼالهخ اُ‪ٜ‬رخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ث‪ٓ ٖ٤‬غٔ‪ٞ‬ػز‪ٓ ٖ٤‬ضَ‬ ‫ػالهخ ‪ٛ‬رخ روِ‪٤‬م‪٣‬خ ٓغ ثؼط اُالرؼ‪. ٖ٤٤‬‬

‫‪ٝ‬‬

‫‪٢ٛ‬‬

‫ػِٔب‬ ‫إ اُؼالهخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُ‪ٜ‬رخ ثٔب رؾز‪ ١ٞ‬ز‪ٝ‬ط روِ‪٤‬م‪ ٓ ١‬رت ‪ٝ‬اً‪٤‬م ٓضَ‬ ‫ثؾ‪٤‬ش إ ∈ ‪ٝ‬‬ ‫إ ∈ ‪ ، ∈ ٝ‬ا‪٣‬عب هم رؾز‪ ١ٞ‬ز‪ٝ‬ط ٓ رت ٓؾزَٔ ٓضَ‬ ‫∈ ‪ٗٝ‬وصم ثبُز‪ٝ‬ط أُ رت أُؾزَٔ ‪ ٞٛ‬اٗ٘ب ؿ‪ٓ ٤‬ز ًم‪ ٖٓ ٖ٣‬آٌبٗ‪٤‬خ ‪ٝ‬ع‪ ٞ‬ػالهخ ث‪ٖ٤‬‬ ‫ٌُٖ ث٘ضجخ ٓؼ‪٘٤‬خ اهَ ٖٓ ‪،100%‬‬ ‫ٖٓ ػمٓ‪ ، ٚ‬ا‪ ٝ‬ثٔب ر‪ٞ‬عم ػالهخ ث‪ٖ٤‬‬ ‫أُ ًجز‪ٖ٤‬‬ ‫كٔضال اُؼالهخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫{‬ ‫{ }‬ ‫}‬ ‫)‪(27‬‬ ‫ٓؼ كخ ػِ‪ٓ ٠‬ز أُغٔ‪ٞ‬ػخ ًٔب ‪٢ِ٣‬‬ ‫{‬ ‫}‬ ‫ثبُز ً‪٤‬م ر٘زٔ‪ ٢‬اُ‪ٝ ٠‬ث٘ضجخ ‪ 100%‬ث‪ٔ٘٤‬ب‬ ‫لذ وٕ االز‪ٝ‬اط أُ رجخ‬ ‫‪٘٣‬زٔ‪ ٢‬اُ‪ٝ r ٠‬ث٘ضجخ ‪ٝ 60%‬ث٘ضجخ ‪ِٓ 10%‬ؾو‪٤‬ز‪ٜ‬ب ؿ‪ٓ ٤‬ؼ‪٘٤‬خ ‪ٝ‬اخ‪ ٤‬ا ك‪ ٢ٜ‬ال‬ ‫اُؼ٘ص‬ ‫ال ٗؼِْ ش‪٤‬ئب ػٖ‬ ‫‪ٝ‬‬ ‫ر٘زٔ‪ ٢‬اُ‪ٗ ٠‬ضجخ ‪ٝ ، 30%‬ك‪ٔ٤‬ب ‪٣‬خا االز‪ٝ‬اط أُ رجخ‬ ‫آٌبٗ‪٤‬خ ِٓؾو‪٤‬ز‪ِ ْٜ‬ع ‪ٝ( r‬هم ‪ٌ٘ٔٓ ٌٕٞ٣‬ب )‪.‬‬ ‫انرؼرَف انؼاو نهؼاللح انُ​ُىذروصىفكُح‬ ‫ث‪ٞ‬صبغخ ا رجبغبد ث‪ٓ ٖ٤‬غبٓ‪٤‬غ عزئ‪٤‬خ ‪ٝ‬ال‬ ‫‪ ٌٖٔ٣‬ص‪٤‬بؿخ اُؼالهخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫رؼ‪٘٤٤‬بد ك‪ٓ ٢‬غ ٓغبٓ‪٤‬غ عزئ‪٤‬خ ‪ٝ‬ال رؼ‪٘٤٤‬بد ك‪. ٢‬‬ ‫إ اُزؼ ‪٣‬ق االخ‪ ٤‬ثٔضبثخ رؼٔ‪ٓ ْ٤‬عبػق ُِؼالهبد اُزوِ‪٤‬م‪٣‬خ ‪،‬‬ ‫ا‪ٝ‬ال ‪ :‬ألٗ‪ٜ‬ب ثمال ٖٓ إ ر ثػ ػ٘بص ك‪ٓ ٢‬غ ػ٘بص ٖٓ ٗغم اٗ‪ٜ‬ب ر ثػ ٓغبٓ‪٤‬غ عزئ‪٤‬خ ٖٓ‬ ‫‪ A‬ثٔغبٓ‪٤‬غ عزئ‪٤‬خ ك‪. B ٢‬‬ ‫صبٗ‪٤‬ب ‪ :‬اٗ‪ٜ‬ب رِٔي ثؼط اُالرؼ‪٘٤٤‬بد ٓز اثطخ ا‪ ٝ‬ثؼط اُ ‪ٝ‬اثػ ال رٌ‪ٓ ٕٞ‬ؼ ‪ٝ‬كخ ‪.‬‬ ‫إ اُؼالهخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٝ ،‬اُز‪ ٢‬ال رٔضَ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٌٖٔ٣ ،‬رو‪٤٤‬م‪ٛ‬ب ثؼمح غ م‬ ‫ُزؾ‪ِٜ٣ٞ‬ب اُ‪ ٠‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪.‬‬ ‫‪ ٌٖٔ٣‬ر‪ٞ‬ؽ‪٤‬م ذُي ًٔب ‪٢ِ٣‬‬ ‫ثؾ‪٤‬ش‬ ‫‪ٝ‬‬ ‫ٓضبٍ ذُي‪ٗ ،‬لز ض وٕ‬ ‫{‬ ‫و‪} ٝ‬‬ ‫‪.‬‬ ‫‪or‬‬ ‫و‪ٝ‬‬ ‫آب ‪and‬‬ ‫‪ٝ‬ث‪ٜ‬اا ٌٗ‪ ٕٞ‬هم رؼ ك٘ب ػِ‪ ٠‬اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪...‬‬

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‫‪Neutrosophic Science International Association‬‬

‫‪ – 4.2‬انذانح انهشح انُ​ُىذروصىفكُح‬ ‫اُماُخ اُ‪ٜ‬رخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪ٝ‬‬ ‫ٓضَ ∈ ٓغ‬ ‫اخزجب أُضزو‪ ْ٤‬اُؼٔ‪ ١ ٞ‬اُزوِ‪٤‬م‪.)١‬‬

‫‪ ٢ٛ‬ػالهخ ‪ٛ‬رخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثؾ‪٤‬ش ُ‪ٝ ٞ‬عم ػ٘ص‬ ‫‪ٛ( .‬اا ‪ٔ٣‬ضَ‬ ‫ػ٘مئا‬ ‫‪ ،‬ػِٔب إ ‪∈ ,‬‬

‫‪ - 5.2‬انذانح انُ​ُىذروصىفكُح انؼايح‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُؼبٓخ ‪ ٢ٛ‬ػالهخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ال ‪٣‬لِؼ ك‪ٜ٤‬ب اخزجب أُضزو‪ ْ٤‬اُؼٔ‪( ١ ٞ‬ا‪ٝ‬‬ ‫اخزجب أُضزو‪ ْ٤‬اُؼٔ‪ ١ ٞ‬ذ‪ ٝ‬أُغٔ‪ٞ‬ػخ اُغزئ‪٤‬خ ) ‪ٝ ،‬ك‪ٛ ٢‬ا‪ ٙ‬اُؾبُخ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُؼبٓخ‬ ‫رزطبثن ٓغ اُؼالهخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪.‬‬

‫‪ -6.2‬انذانح انُ​ُىذروصىفكُح انهشح (أٌ انرٍ يجانها ويجانها انًماتم يجايُغ‬ ‫جزئُح)‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ( اُ‪ٜ‬رخ ا‪ ٝ‬اُز‪ٓ ٢‬غبُ‪ٜ‬ب ‪ٓٝ‬غبُ‪ ٚ‬أُوبثَ ٓغبٓ‪٤‬غ عزئ‪٤‬خ) ‪ ٢ٛ‬ثبُؼٔ‪ ّٞ‬اُخ‬ ‫رِٔي ثؼط اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م ) ‪ ٖٓٝ .‬االٓضِخ ػِ‪ ٠‬ذُي‬ ‫‪ ُ٘ -1‬خا ‪ٛ‬ا‪ ٙ‬اُماُخ‬ ‫{‬ ‫{ }‬ ‫}‬ ‫)‪(28‬‬ ‫(ٗؾٖ ؿ‪ٓ ٤‬ز ًم‪)ٖ٣‬‬ ‫ٌُٖ ا‪ٝ‬‬ ‫ُ‪ ٞ‬رْ رٔض‪ َ٤‬اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اػال‪ ٙ‬ثٔخطػ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ، ٢‬ص‪ٌٕٞ٤‬‬

‫ٓخطػ (‪ )1‬رٔض‪ َ٤‬ثٔخطػ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫إ االص‪ ْٜ‬أُ٘وطخ ك‪ ٢‬أُخطػ اُضبثن رمٍ ػِ‪ ٠‬اٗ٘ب ؿ‪ٓ ٤‬ز ًم‪ ٖ٣‬ك‪ٔ٤‬ب اذا ًبٕ اُؼ٘ص ‪ 3‬ك‪٢‬‬ ‫أُغبٍ ٓ رجػ ثبُؼ٘ص ‪ 6‬ا‪ ٝ‬اُؼ٘ص ‪ ٖٓ 7‬أُغبٍ أُوبثَ‪.‬‬ ‫ًٔب ٗ ‪ ، ٟ‬إ ‪ٛ‬ا‪ ٙ‬اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ال رٔضَ اُخ ثبُٔل‪ ّٜٞ‬اُزوِ‪٤‬م‪ ١‬أُزؼب ف ػِ‪ً ٚ٤‬اُي ‪٢ٛ‬‬ ‫ال رٔضَ ػالهخ ثبُط ‪٣‬وخ اُزوِ‪٤‬م‪٣‬خ‪.‬‬ ‫االٕ ُ‪ ٞ‬ا ٗب ػَٔ ٓغٔ‪ٞ‬ػخ رٔضَ ‪ٛ‬ا‪ ٙ‬اُمُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ص‪ُ ٌٕٞ٤‬م‪٘٣‬ب‪.‬‬ ‫{ ؽ‪٤‬ش إ اُو‪ ْ٤‬اخَ اُؾم‪ ٝ‬أُ٘وطخ رؼ٘‪ ٢‬اٗ٘ب ؿ‪٤‬‬ ‫}‬

‫‪22‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ٓز ًم‪ ٖٓ ٖ٣‬إ ‪ٛ‬ا‪ ٙ‬اُؼ٘بص ر٘زٔ‪ ٢‬اُ‪ٛ ٠‬ا‪ ٙ‬أُغٔ‪ٞ‬ػخ اّ ال‪٘​ٌ٘ٔ٣ٝ .‬ب ‪ٝ‬ظغ االز‪ٝ‬اط )‪ٝ (3, 6‬‬ ‫)‪ (3, 7‬ثِ‪ ٕٞ‬اؽٔ ُالٗزجب‪ٝ ٙ‬ثزٔض‪ٛ َ٤‬ا‪ ٙ‬اُماُخ ك‪ ٢‬عم‪ ٍٝ‬ص٘ؾصَ ػِ‪ٓ ٠‬ب ‪-:٢ِ٣‬‬

‫عم‪)1( ٍٝ‬‬ ‫‪ ُٞٝ‬ؽب‪ُ٘ٝ‬ب رٔض‪ َ٤‬اُماُخ ث‪٤‬بٗ‪٤‬ب ص‪ ٌٕٞ٤‬االر‪-:٢‬‬

‫اُ صْ اُج‪٤‬بٗ‪)1( ٢‬‬ ‫‪ ُٞٝ‬غ‪ٗ ٞ‬ب هِ‪٤‬ال ‪ٛ‬اا أُضبٍ ثبُط ‪٣‬وخ االر‪٤‬خ ‪ ٢ٛٝ‬وٕ اُو‪ٔ٤‬خ ‪ ٓ 3‬رجطخ ٓغ ‪ 7‬عزئ‪٤‬ب ُ٘وَ‬ ‫)‪.(3, 7)(0.6, 0.2, 0.5‬‬ ‫ٓٔب ‪٣‬ؼ٘‪ ٢‬وٕ ‪ ٓ 3‬رجطخ ثع ‪ 7‬ث٘ضجخ ‪ 60%‬آب ٗضجخ ػمّ ‪ٝ‬ظ‪ٞ‬ػ اال رجبغ ٖٓ ػمٓ‪ ٚ‬كجِؾ ‪20%‬‬ ‫‪ٗٝ‬الؽع إ ‪ 3‬ؿ‪ ٓ ٤‬رجطخ ثع ‪ 7‬ث٘ضجخ ‪. 50%‬‬ ‫إ ٓغٔ‪ٞ‬ع ‪ 0.6 + 0.2 + 0.5 = 1.3‬اًج ٖٓ ‪ 1‬الٗ‪ُ ٚ‬م‪٘٣‬ب صالس ٓصب رغ‪ٜ‬زٗب ثبُٔؼِ‪ٓٞ‬بد‬ ‫ؽ‪ ٍٞ‬ر ً‪٤‬م اُز اثػ و‪ ٝ‬اُالرؼ‪ ٖ٤٤‬ك‪ ٢‬اُز اثػ ٖٓ ػمٓ‪ٝ ٚ‬ثرٌَ ٓضزوَ ثبصزخماّ ٓو‪٤‬بس ٓخزِق‬ ‫ُِؾضبة‪.‬‬ ‫‪ -1‬ص٘ط‪ ٓ ٞ‬ح اخ ‪ٛ ٟ‬ا‪ ٙ‬اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ًٔٝ‬ب ‪٢ِ٣‬‬ ‫}‬

‫)‪(29‬‬ ‫‪and 7.‬‬

‫{‬

‫}‬

‫{‬

‫‪, but‬‬

‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ُ‪٤‬ضذ اُخ رم س ثط ‪٣‬وخ ًالص‪٤ٌ٤‬خ (روِ‪٤‬م‪٣‬خ) ‪( ،‬ألٗ‪ ٚ‬ػ٘م ‪ٛ‬ا‪ ٙ‬اُماُخ ‪٣‬لرَ‬ ‫) ‪ ،‬لٕ ُِماُخ ‪ g‬ا ثغ ر اثطبد (رٔض‪٤‬الد) ًٔب ك‪ ٢‬أُخطػ‬ ‫اخزجب أُضزو‪ ْ٤‬اُؼٔ‪ ١ ٞ‬ػ٘م‬ ‫االَر‪٢‬‬

‫‪23‬‬

‫‪Neutrosophic Science International Association‬‬

‫ٓخطػ (‪)1‬‬ ‫{‬

‫}‬

‫عم‪)1( ٍٝ‬‬

‫اُ صْ اُج‪٤‬بٗ‪)1( ٢‬‬ ‫ٓ ح اخ ‪ُٞ ، ٟ‬وػمٗب رصٔ‪ ْ٤‬اُماُخ ‪ًٔ g‬ب ‪٢ِ٣‬‬ ‫)‪(30‬‬ ‫ثاُي‬

‫‪} ,‬‬

‫} {‬ ‫‪, and‬‬ ‫رصجؼ اُخ روِ‪٤‬م‪٣‬خ ذ‪ٓ ٝ‬غبٍ ٓوبثَ ثرٌَ ٓغٔ‪ٞ‬ػخ ‪.‬‬

‫{‬

‫}‬ ‫‪,‬‬

‫{‬

‫‪ -1‬ػ٘ب ػز‪٣‬ز‪ ١‬اُوب ئ ٗزط م اُ‪ ٠‬غ از ٓخزِق ٖٓ اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫)‪(31‬‬ ‫∈ ‪, for any‬‬

‫∈‬ ‫‪24‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ُاا كبٕ اُؾو‪٤‬وخ اُ‪ٞ‬ؽ‪٤‬مح اُز‪ٗ ٢‬ؼِٔ‪ٜ‬ب ؽ‪ٛ ٍٞ‬ا‪ ٙ‬اُماُخ اٗ‪ٜ‬ب ٓو‪٤‬مح ثبُٔضزو‪ ٖ٤ٔ٤‬االكو‪ٖ٤٤‬‬

‫‪ٝ‬‬

‫اُ صْ اُج‪٤‬بٗ‪)1( ٢‬‬ ‫‪ -4‬ثط ‪٣‬وخ ٓربث‪ٗ ، ٜٚ‬ضزط‪٤‬غ رط‪٣ٞ‬‬ ‫اُضٔي)‬

‫ُ٘ؾصَ ػِ‪ ٠‬اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُضبثزخ ( ا‪ ٝ‬اُخ‬

‫)‪(32‬‬ ‫رٔضَ ٓغٔ‪ٞ‬ػخ ًَ أُغبٓ‪٤‬غ اُغزئ‪٤‬خ ٖٓ‬ ‫ؽ‪٤‬ش‬ ‫اُؼٔ‪ُِٔ ١ ٞ‬ضزو‪. [2,3] ْ٤‬‬

‫‪for any ∈ ,‬‬ ‫‪ ٞٛ‬ذُي أُوطغ اُر ‪٣‬ػ‬ ‫‪ ،‬كٔضال‬

‫اُ صْ اُج‪٤‬بٗ‪)4( ٢‬‬ ‫‪ -1‬اُخ اُضٔي اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ؿ‪ ٤‬اُضبثزخ (أُزـ‪ ٤‬ح)‬ ‫)‪(33‬‬ ‫‪25‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ٝ‬شٌَ ‪ٛ‬ا‪ ٙ‬اُماُخ ‪ٞٛ‬‬

‫كٔضال‬

‫اُ صْ اُج‪٤‬بٗ‪)1( ٢‬‬ ‫‪.‬‬

‫‪ -6‬ػٔ‪ٓٞ‬ب ‪ٗ ،‬ضزط‪٤‬غ رؼ ‪٣‬ق اُخ اُضٔي اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ًٔب ‪٢ِ٣‬‬ ‫)‪(34‬‬

‫اُ صْ اُج‪٤‬بٗ‪)6( ٢‬‬ ‫‪ٝ‬ثبُزبُ‪ ٢‬كنٕ ه‪ٞ‬ص‪ ٢‬اُلز ح أُـِو‪ ٖ٤‬هم ‪ٌٗٞ٣‬بٕ كز ح ٓلز‪ٞ‬ؽخ ‪),‬‬ ‫‪, or‬‬ ‫ٓلز‪ٞ‬ؽخ (شج‪ٓ ٚ‬ـِوخ)‬

‫( ا‪ ٝ‬كز اد شج‪ٚ‬‬ ‫‪،‬‬

‫‪26‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ثاُي ٗضزط‪٤‬غ رؾم‪٣‬م اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫خا ٓضال‬ ‫‪ٝ‬اُر ‪٣‬ػ اُؼٔ‪ٓ ٖٓ ١ ٞ‬ضزو‪ .ْ٤‬إ االٓضِخ اُضبثوخ ُم‪ٝ‬اٍ اُضٔي اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٢ٛ‬ك‪ ٢‬ؽو‪٤‬وز‪ٜ‬ب‬ ‫صط‪ٞ‬ػ روِ‪٤‬م‪٣‬خ ك‪ ٢‬اُلعبء ‪. 2‬‬ ‫‪ٓ -1‬ضبٍ ػٖ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثغزئ‪ٖ٤‬‬ ‫)‪(35‬‬ ‫{‬ ‫الؽع اُ صْ اُج‪٤‬بٗ‪ ٢‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ُٜ ٢‬ا‪ ٙ‬اُماُخ ‪.‬‬

‫اُ صْ اُج‪٤‬بٗ‪)1( ٢‬‬ ‫‪ٔ٣‬ضَ ذُي أُوطغ (اُر ‪٣‬ػ) اُؼٔ‪ ١ ٞ‬أُـِن‬

‫‪ُِٔٝ‬ز‪٣‬م الؽع إ‬ ‫ُِٔضزو‪.[AB] ْ٤‬‬ ‫‪ٝ‬ك‪ ًَ ٢‬االٓضِخ اُضبثوخ ٗغم إ اُالرؼ‪ ٖ٤٤‬هم ظ‪ ٜ‬ك‪ ٢‬ه‪ ْ٤‬اُماُخ‪ٝ .‬ثطج‪٤‬ؼخ اُؾبٍ ‪ ٌٖٔ٣‬إ ‪٣‬ظ‪ٜ‬‬ ‫اُالرؼ‪ ٖ٤٤‬ك‪ٓ ٢‬غبٍ اُماُخ ا‪ ٝ‬ك‪ً ٢‬ال ٓغبُ‪ٜ‬ب ‪ٓٝ‬غبُ‪ٜ‬ب أُوبثَ ًٔب ‪ٓ ٞٛ‬الؽع ك‪ٔ٤‬ب ‪.٢ِ٣‬‬ ‫‪ -2‬اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ٖٓ خالٍ ٓغبٍ اُماُخ‬ ‫)‪(36‬‬ ‫‪٘ٛ‬ب ال ٗؼِْ اذا ًبٗذ‬ ‫‪ٓٝ‬ضبٍ اخ ‪ٞٛ‬‬ ‫)‪(37‬‬

‫و‪ٝ‬‬

‫{‬

‫}‬

‫{‬

‫}‬

‫‪.‬‬ ‫}‬

‫{‬

‫}‬

‫{‬

‫‪ -3‬اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ك‪ً ٢‬ال أُغبٍ ‪ٝ‬أُغبٍ أُوبثَ‪.‬‬ ‫)‪(38‬‬

‫}‬

‫{‬

‫}‬

‫{‬

‫‪ٝ‬اُا‪٣ ١‬ؼ٘‪٢‬‬ ‫‪27‬‬

‫‪Neutrosophic Science International Association‬‬ ‫;‪.U(1) = 5, or U(1) = 6, or U(1) = 7, or U(2) = 5, or U(2) = 6, or U(2) = 7‬‬ ‫ٓضبٍ اخ‬ ‫‪,‬‬ ‫)‪(39‬‬ ‫لٕ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ االخ‪ ٤‬ح ثبُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ك‪ٓ ٢‬غبُ‪ٜ‬ب ‪ ٌٖٔ٣‬إ رؤ‪ ٍٝ‬ثط ‪٣‬وخ‬ ‫‪ ٌٖٔ٣‬ر عٔز‪ٜ‬ب‬ ‫ثاُي كبٕ اُماُخ‬ ‫رٌبكئ‬ ‫اخ ‪ٗ : ٟ‬الؽع إ‬ ‫‪ .‬إ ‪ٛ‬ا‪ٙ‬‬ ‫اُ‪ ٠‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثالرؼ‪ ٖ٤٤‬ك‪ٓ ٢‬غبُ‪ٜ‬ب أُوبثَ كوػ‪،‬‬ ‫اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ االخ‪ ٤‬ح ال رٔضَ ػالهبد ثبُط ‪٣‬وخ اُزوِ‪٤‬م‪٣‬خ ‪.‬‬

‫‪ -2.2‬انالذؼُ​ٍُ (انالذحذَذ) انًرمطغ وغُر انًرمطغ‬ ‫ٖٓ ‪ٝ‬ع‪ٜ‬خ ٗظ اخ ‪٘ٛ ، ٟ‬بى الرؼ‪ٓ ٖ٤٤‬زوطغ ‪ٛ‬اا ٓب ٗغم‪ ٙ‬ك‪ ٢‬االٓضِخ االر‪٤‬خ‪:‬‬ ‫‪,‬‬ ‫‪,‬‬

‫‪or‬‬ ‫‪or‬‬

‫‪ٝ‬الرؼ‪ ٖ٤٤‬ؿ‪ٓ ٤‬زوطغ ‪ًٔ ،‬ب ٗغم‪ ٙ‬ك‪ٔ٤‬ب ‪-:٢ِ٣‬‬ ‫‪or‬‬ ‫‪or‬‬ ‫‪or‬‬ ‫‪.‬‬ ‫‪ٝ‬ثبالػزٔب ػِ‪ٞٗ ًَ ٠‬ع ٖٓ اٗ‪ٞ‬اع اُالرؼ‪ٗ ٖ٤٤‬ؾٖ ثؾبعخ اُ‪ ٠‬رؾم‪٣‬م رو٘‪٤‬خ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ خبصخ‬ ‫رٌٔ٘​٘ب ٖٓ رغب‪ٝ‬ز ‪ٛ‬اا اُالرؼ‪. ٖ٤٤‬‬

‫‪ -2.2‬دوال َُىذروصىفكُح تمُى يرجهح راخ يرغُراخ يرؼذدج‬ ‫ك‪ ٢‬اُج٘‪ ٞ‬اُضبثوخ رْ اػطبء آضِخ ُم‪ٝ‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثو‪ٓٝ ْ٤‬زـ‪ ٤‬اد ؽو‪٤‬و‪٤‬خ ‪ُ٘ٝ‬لش اُم‪ٝ‬اٍ اُز‪٢‬‬ ‫ٗبهر٘ب‪ٛ‬ب ‪ٞ٣‬عم ك‪ ٢‬ا‪ٓ ١‬غبٍ ػِٔ‪ٝ ٢‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثو‪ٓ ْ٤‬زغ‪ٜ‬خ ذاد ٓزـ‪ ٤‬اد ٓزؼم ح ‪.‬‬

‫)‪(40‬‬

‫)‬

‫(‬

‫‪ ٌٖٔ٣‬إ رٌ‪ ٕٞ‬كعبءاد ػِٔ‪٤‬خ ٖٓ و‪ٞٗ ١‬ع‪ .‬إ‬ ‫‪ٝ‬‬ ‫ثبُز ً‪٤‬م‬ ‫‪ٌٛ‬اا ‪ٝ‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ذاد اُو‪ ْ٤‬أُزغ‪ٜ‬خ ثٔزـ‪ ٤‬اد ػم‪٣‬مح ‪ ٌٖٔ٣‬وٕ رِٔي الرؼ‪ ٖ٤٤‬ك‪ٓ ٢‬غبُ‪ٜ‬ب ا‪ٝ‬‬ ‫ك‪ٓ ٢‬غبُ‪ٜ‬ب أُوبثَ ا‪ٜٔ٤ًِ ٝ‬ب ٓؼب ‪ٝ ،‬اُالرؼ‪ ٌٖٔ٣ ٖ٤٤‬إ ‪ٓ ٌٕٞ٣‬زوطؼب ا‪ ٝ‬ؿ‪ٓ ٤‬زوطغ‪.‬‬

‫‪28‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ -2.2‬انذوال انضًُ​ُح انُ​ُىذروصىفكُح‬ ‫‪ٝ‬ث٘لش اُط ‪٣‬وخ اُز‪ً٘ ٢‬ب هم ٗؼ ك٘ب ك‪ٜ٤‬ب ػِ‪ ٠‬اُم‪ٝ‬اٍ اُص ‪٣‬ؾخ ‪ٝ‬اُم‪ٝ‬اٍ اُعٔ٘‪٤‬خ ك‪ ٢‬أُ٘طن‬ ‫اُزوِ‪٤‬م‪( ١‬اٌُالص‪ :)٢ٌ٤‬ر‪ٞ‬عم ‪ٝ‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ص ‪٣‬ؾخ ٓضَ‬ ‫‪or‬‬ ‫)‪(41‬‬ ‫‪ًٝ‬اُي ‪ٝ‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ظٔ٘‪٤‬خ ‪ٓ ،‬ضَ‬ ‫}‬

‫)‪(42‬‬

‫|‬

‫∈‬

‫{‬

‫‪ -11.2‬ذركُة انذوال انُ​ُىذروصىفكُح‬ ‫إ ر ً‪٤‬ت اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٔ٣‬ضَ ر‪ٞ‬ص‪٤‬ؼب ُز ً‪٤‬ت اُم‪ٝ‬اٍ أُ‪ٞ‬ع‪ ٞ‬ح ك‪ ٢‬أُ٘طن اُزوِ‪٤‬م‪ٓ ١‬غ‬ ‫كب م ‪ٝ‬ع‪ ٞ‬اُالرؼ‪ً ٖ٤٤‬زؼٔ‪ٓ .ْ٤٤‬ضبٍ ُزٌٖ‬ ‫‪for‬‬ ‫)‪(43‬‬ ‫‪ٝ‬‬ ‫{‬

‫)‪(44‬‬ ‫اُز‪ٞ٤ٗ ٖ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬ز‪ٓ ، ٖ٤‬ب ‪ ٢ٛ‬اُماُخ‬ ‫)‬

‫(‬

‫)‪(45‬‬ ‫ٗغم إ اُالرؼ‪ ٖ٤٤‬أُزوطغ " ‪ 7‬ا‪ٓ"9 ٝ‬غ اُالرؼ‪ ٖ٤٤‬أُضزٔ (ؿ‪ ٤‬أُزوطغ ) ك‪٢‬‬ ‫“ رْ ثض‪ٜٔ‬ب ا‪ٗ ٝ‬ر ‪ٔٛ‬ب ك‪ ٢‬اُالرؼ‪ ٖ٤٤‬أُضزٔ (ؿ‪ ٤‬أُزوطغ) أُز ‪ٝ‬ط‬ ‫”‬ ‫“‬ ‫‪or‬‬ ‫”‬ ‫ٖٓ ٗبؽ‪٤‬خ اخ ‪ ٞٗ ٟ‬إ ٗط ػ اُضؤاٍ االر‪ٓ : ٢‬ب ‪ ٢ٛ‬ه‪ٔ٤‬خ‬ ‫*‬

‫‪+‬‬

‫(‬

‫)‬

‫)‪(46‬‬ ‫ٓب ‪ ٢ٛ‬اُص‪٤‬ـخ اُؼبٓخ‬ ‫)‬

‫)‪(47‬‬

‫‪)+‬‬ ‫]‬

‫(‬ ‫(‬

‫{‬

‫)‬

‫)‬

‫( *‬

‫(‬

‫[‬

‫{‬

‫‪29‬‬

‫‪Neutrosophic Science International Association‬‬ ‫الٕ ٓغبٍ اُماُخ‬

‫‪ٞٛ‬‬

‫كبٗ‪ُ ٚ‬م‪٘٣‬ب‬

‫‪ٛ‬اا ثبُ٘ضجخ ُِغزء اال‪ٖٓ ٍٝ‬‬

‫ا‪١‬‬

‫هم رِٔي الرؼ‪٘٤٤‬ب ك‪ٓ ٢‬غبُ‪ٜ‬ب ا‪ٝ‬‬ ‫‪ًٔٝ .‬ب ذً ٗب صبثوب ‪ ،‬اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫اُماُخ‬ ‫ٓغبُ‪ٜ‬ب أُوبثَ ا‪ ٝ‬ك‪ ٢‬اُؼالهخ ث‪( ٝ ٖ٤‬ا‪ ٝ‬أل‪ ١‬ؽبُز‪ ٖ٤‬ا‪ ٝ‬صالس ؽبالد ٖٓ ذُي ٓؼب)‪.‬‬

‫‪ - 11.2‬يؼكىس انذانح انُ​ُىذروصىفكُح‬ ‫إ ٓؼٌ‪ٞ‬س اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٞٛ‬ا‪٣‬عب اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٝ‬ذُي الٕ اُالرؼ‪ ٖ٤٤‬أُ‪ٞ‬ع‪ ٞ‬ك‪٢‬‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ االصِ‪٤‬خ ‪٣‬زْ ر‪٣ ٞ‬ض‪ُٔ ٚ‬ؼٌ‪ٞ‬س رِي اُماُخ ‪.‬‬ ‫{‬

‫)‪(48‬‬ ‫ا‪ٝ‬‬

‫‪0≠x‬‬ ‫‪0‬‬

‫;‪2x+1 or 6x‬‬ ‫‪[1, 3].‬‬ ‫ػ٘ب ٗغم ٓؼٌ‪ٞ‬س اُماُخ‬ ‫)‪(49‬‬ ‫‪or‬‬ ‫ثٔب إ ‪,‬‬

‫‪ًٔٝ‬ب ‪، ٢ِ٣‬‬ ‫ٗؼ‪٤‬م ًزبثخ اُماُخ ث ثماٍ أُزـ‪ ٤‬اد‬

‫ُغٔ‪٤‬غ ه‪ْ٤‬‬

‫أُضزوِخ ُ٘غؼِ‪ٜ‬ب ٓزـ‪ ٤‬اد ٓؼزٔمح ‪ٝ‬ثبُؼٌش ‪ًٔٝ‬ب ‪٢ِ٣‬‬ ‫ٓٔب ‪٣‬ؤ ‪ ١‬اُ‪٠‬‬

‫ًاُي ُم‪٘٣‬ب‬

‫‪ٝ‬ثبُزبُ‪ ٢‬كبٕ‬

‫‪ٝ‬ثبُزبُ‪ ٢‬كبٕ‬

‫‪.‬‬ ‫‪ٌٛٝ‬اا كبٕ ٓؼٌ‪ٞ‬س اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬

‫ٓٔب ‪٣‬ؤ ‪ ١‬اُ‪٠‬‬

‫‪: ٢ٛ‬‬ ‫{‬

‫)‪(50‬‬ ‫ٓ ح اخ ‪ٗ ٟ‬غم إ ٓؼٌ‪ٞ‬س اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫∈‬

‫‪Or‬‬ ‫ثجضبغخ أُؼٌ‪ٞ‬س ‪ٞٛ‬‬ ‫‪,‬‬

‫∈ ‪, for all‬‬

‫)‪(51‬‬ ‫كبٕ ٓؼٌ‪ٞ‬ص‪ٜ‬ب ‪ٞٛ‬‬ ‫‪or‬‬ ‫آب ثبُ٘ضجخ ُِماُخ االص‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُزبُ‪٤‬خ‬ ‫‪.‬‬ ‫)‪(52‬‬ ‫‪ٝ‬ثبُط ‪٣‬وخ ٗلض‪ٜ‬ب كبٕ ٓؼٌ‪ٞ‬س اُماُخ اُِ‪ٞ‬ؿب ‪٣‬زٔ‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُزبُ‪٤‬خ ‪:‬‬ ‫)‪(53‬‬

‫‪or‬‬

‫‪.‬‬ ‫‪31‬‬

‫‪Neutrosophic Science International Association‬‬ ‫اُماُخ اُزوِ‪٤‬م‪٣‬خ رٌ‪ ٕٞ‬هبثِخ ُالٗؼٌبس اذا ‪ٝ‬كوػ اذا ًبٗذ رؾون ش غ اُزوبثَ ‪ٝ‬اؽم ُ‪ٞ‬اؽم (ا‪ ١‬إ‬ ‫اُماُخ رؾون اخزجب أُضزو‪ ْ٤‬اُؼٔ‪.)١ ٞ‬‬ ‫ػ٘ب ٗ خا ثبالػزجب اُماُخ اُزوِ‪٤‬م‪٣‬خ اُزبُ‪٤‬خ‬ ‫{‬ ‫} { }‬ ‫)‪(54‬‬ ‫ٗالؽع إ ‪ٛ‬ا‪ ٙ‬اُماُخ ال رؾون روبثَ ‪ٝ‬اؽم ٍ ‪ٝ‬اؽم الٗ‪ٜ‬ب رلرَ ك‪ ٢‬اخزجب أُضزو‪ ْ٤‬اُؼٔ‪ ١ ٞ‬ػ٘م‬ ‫‪ُ .‬اُي كبٕ ‪ٛ‬ا‪ ٙ‬اُماُخ ؿ‪ ٤‬هبثِخ ُالٗؼٌبس ثرٌَ روِ‪٤‬م‪.١‬‬ ‫ؽ‪٤‬ش إ‬ ‫‪,‬‬ ‫ػِ‪ ٠‬ا‪٣‬خ ؽبٍ ‪ٞ٤ٗ ،‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ب ‪٘​ٌ٘ٔ٣ ،‬ب اخا أُؼٌ‪ٞ‬س اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ُِ ٢‬ماُخ ‪ًٔٝ‬ب ‪.٢ِ٣‬‬ ‫} {‬ ‫} {‬ ‫{‬ ‫}‬ ‫)‪(55‬‬ ‫ٗؾٖ ثؾبعخ الٕ ٗ خا ٓوِ‪ٞ‬ة اُماُخ ‪ ٕٝ‬إ‬ ‫ٖٓ اعَ ٓؼٌ‪ٞ‬س اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪ ،‬لٕ اُالرؼ‪ٖ٤٤‬‬ ‫اُخبص ث صْ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ٗ٘ض‪ ٠‬أُؾ‪ ٞ‬اُز٘بظ ‪١‬‬ ‫(اُالرؾم‪٣‬م) ُِماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ص‪٤‬زْ ر‪٣ ٞ‬ض‪ُٔ ٚ‬ؼٌ‪ٞ‬س رِي اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪.‬‬

‫يضأنح يمررحح‬ ‫و‪ ١‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٢ٛ‬اُخ هبثِخ ُالٗؼٌبس‬ ‫اُج ‪ٛ‬بٕ‬ ‫كبٗ‪ٖٓٝ ٚ‬‬ ‫ؽ‪٤‬ش‬ ‫ك‪ ٢‬اخزجب أُضزو‪ ْ٤‬اُؼٔ‪ ١ ٞ‬ػ٘م‬ ‫اذا كرِذ اُماُخ‬ ‫رؼ ‪٣‬ق أُغبٍ ُِماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ص٘غم إ ٓؼٌ‪ٞ‬س اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫∈ {‬ ‫}‬ ‫)‪(56‬‬ ‫اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ اذا ًبٗذ اُ صٔخ اُج‪٤‬بٗ‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ِع‬ ‫ٓ ح اخ ‪ُ ، ٟ‬زٌٖ‬ ‫ثؾ‪٤‬ش إ ‪ ⊆ ٝ ⊆ ,‬ػ٘مئا كبٕ اُ صْ اُج‪٤‬بٗ‪ُٔ ٢‬ؼٌ‪ٞ‬س ‪ٛ‬ا‪ٙ‬‬ ‫رؾز‪ ١ٞ‬اُ٘وطخ‬ ‫‪ .‬إ ا‪ٗ ١‬وطخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ص‪٤‬ؾز‪ٗ ١ٞ‬وطخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪ ،‬ؽ‪٤‬ش إ ∈ ‪ٝ ∈ ٝ‬اُز‪ ٢‬ثؼم‪ٛ‬ب ‪٣‬ضب‪ ١ٝ‬صل ‪.‬‬ ‫رؼزج رؼٔ‪ُِ٘ ْ٤‬وطخ اُزوِ‪٤‬م‪٣‬خ‬ ‫إ اُ٘وطخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثبُؼٔ‪ٗ ٢ٛ ّٞ‬وطخ ذاد صٔي ‪ٝ‬اُز‪ ٢‬ثٔب رِٔي ثؼما ثٔوما ‪ 0, 1, 2‬ا‪ٝ‬‬ ‫اًض ( ‪ٛ‬اا ‪٣‬ؼزٔم ػِ‪ ٠‬اُلعبء اُا‪ٗ ١‬ؼَٔ ك‪.)ٚ٤‬‬ ‫ذاد اُجؼم ‪2‬‬ ‫‪ًٝ‬بٓضِخ ‪ُ ،‬م‪٘٣‬ب اُ٘وطخ‬

‫‪30‬‬

Neutrosophic Science International Association

)2 ( ٢ٗ‫ب‬٤‫اُ صْ اُج‬ . ‫ب‬ٛ‫ثؼم‬ٝ

(

) ‫ اُ٘وطخ‬ٝ‫ا‬

)3( ٢ٗ‫ب‬٤‫اُ صْ اُج‬ .0 ٞٛ ‫ب ثؼم‬ِٜ‫ك‬

)12( ٢ٗ‫ب‬٤‫اُ صْ اُج‬

32

{

} ‫آب اُ٘وطخ‬

‫‪Neutrosophic Science International Association‬‬ ‫ُ‪ٜ‬ب ثؼم ‪٣‬ضب‪.3 ١ٝ‬‬

‫ث‪ٔ٘٤‬ب‬

‫اُ صْ اُج‪٤‬بٗ‪)11( ٢‬‬ ‫‪ٓ ‬الؽظخ خبصخ ثبُٔز عٔ‪ٖ٤‬‬ ‫إ اُل م ث‪ ٖ٤‬اُ٘وبغ‬ ‫ُ‪ٜ‬ب صٔي ػِ‪ ٠‬أُؾ‪ٞ‬‬

‫‪ٞٛ‬‬ ‫‪ٝ‬صٔي اخ ػِ‪ ٠‬أُؾ‪ٞ‬‬

‫ُ‪ٜ‬ب صٔي ‪ٝ‬اؽم كوػ ػِ‪ ٠‬أُؾ‪ٞ‬‬

‫ُاا ًبٕ ثؼم‪ٛ‬ب =‪2‬‬

‫ُاا ًبٕ ثؼم‪ٛ‬ب =‪1‬‬

‫ُ‪٤‬ش ُ‪ٜ‬ب صٔي ػِ‪ ٠‬ا‪ٓ ١‬ؾ‪ ٖٓ ٞ‬أُؾب‪ُ ٝ‬اا ًبٕ ثؼم‪ٛ‬ب = ‪0‬‬ ‫ُ‪ٜ‬ب صٔي ػِ‪ ًَ ٠‬أُؾب‪ٝ‬‬

‫ُاا ًبٕ ثؼم‪ٛ‬ب =‪3‬‬

‫‪33‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ - 12.2‬أصفار انذانح انُ​ُىذروصىفكُح‬ ‫اك ض إ‬ ‫ٓغٔ‪ٞ‬ػخ ٓضَ‬

‫⊆‬

‫ثٔب ‪ٝ‬ثرٌَ ػبّ ‪ ٌٕٞ٣‬ثرٌَ‬

‫‪ ،‬لٕ وصلب اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪.‬‬ ‫ثؾ‪٤‬ش إ‬

‫‪ٝ‬ػِ‪ ٠‬صج‪ َ٤‬أُضبٍ‬

‫)‪(57‬‬

‫{‬

‫‪.‬‬

‫‪ٛ‬ا‪ ٙ‬اُماُخ ُ‪ٜ‬ب صل ‪ٛ‬ش إ صل ‪ٛ‬ا‪ ٙ‬اُماُخ ‪ٞٛ‬‬ ‫‪ُٜٝ‬ب صل ثرٌَ كز ح )‪ٞٛ (interval-zero‬‬

‫‪ٝ‬ذُي الٕ‬ ‫الٕ‬

‫‪.‬‬

‫‪ٓ ‬الؽظخ خبصخ ثبُٔز عٔ‪ٖ٤‬‬ ‫‪٣‬وصم ثبُصل اُ‪ٜ‬ش ‪ ٞٛ‬و‪ ١‬ه‪ٔ٤‬خ ػم ‪٣‬خ ٓزؼب ف ػِ‪ٜ٤‬ب ك‪ ٢‬أُ٘طن اٌُالص‪ٝ ٢ٌ٤‬رؤ ‪ ١‬اُ‪ ٠‬عؼَ‬ ‫ص‪ ٞ‬ح اُماُخ = صل‬

‫‪ -13.2‬الذحذَذاخ انذانح‬ ‫ٖٓ االك اغ إ ٗو‪ ٍٞ‬ثبٕ و‪ ١‬اُخ روِ‪٤‬م‪٣‬خ ‪ ٢ٛ‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٌُٖ ،‬ك‪ٛ ٢‬ا‪ ٙ‬اُؾبُخ ٗغم إ‬ ‫اُالرؼ‪ ٖ٤٤‬ك‪ ٢‬اُماُخ اُزوِ‪٤‬م‪٣‬خ ‪ ٌٕٞ٣‬ربك‪ٜ‬ب و‪ٓ ١‬ؼم‪ٓٝ‬ب (‪.)null indeterminacy‬‬

‫‪ -14.2‬انذانح انُ​ُىذروصىفكُح انزوجُح‬ ‫ُ‪ٜ‬ب رؼ ‪٣‬ق ٓربث‪ُِ ٚ‬ماُخ اُز‪ٝ‬ع‪٤‬خ اُزوِ‪٤‬م‪٣‬خ‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُز‪ٝ‬ع‪٤‬خ‬ ‫‪,‬‬ ‫)‪(58‬‬ ‫ؽ‪٤‬ش رٔضَ ٓ ًجخ‬ ‫ُغٔ‪٤‬غ ه‪ ْ٤‬ك‪ٓ ٢‬غ االخا ث٘ظ االػزجب اُزؼٔ‪ ْ٤‬اُزبُ‪٢‬‬ ‫اُالرؼ‪. ٖ٤٤‬‬ ‫ٓضبٍ ‪:‬‬ ‫{‬ ‫}‬ ‫)‪(59‬‬ ‫{‬ ‫‪ٝ‬ثطج‪٤‬ؼخ اُؾبٍ كبٗ‪ ٚ‬ك‪ ٢‬اُغزء أُؼ‪ُِ ٖ٤‬ماُخ ‪ُ ٌٕٞ٣‬م‪٘٣‬ب ٓب ‪٢ِ٣‬‬ ‫)‪(60‬‬ ‫ث‪ٔ٘٤‬ب ك‪ ٢‬اُغزء اُـ‪ٓ ٤‬ؼ‪ُ ٖ٤‬م‪٘٣‬ب ‪or 1‬‬ ‫ٗالؽع‬

‫}‬

‫{‬

‫∈‬

‫ثبُزبُ‪ ٢‬كبٕ‬ ‫‪ٝ‬‬ ‫‪,‬‬ ‫‪34‬‬

‫‪Neutrosophic Science International Association‬‬ ‫‪ٌٛٝ‬اا كبٕ‬

‫‪ ٢ٛ‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ز‪ٝ‬ع‪٤‬خ ‪.‬‬

‫اُ صْ اُج‪٤‬بٗ‪)11( ٢‬‬ ‫إ ٓخطػ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُز‪ٝ‬ع‪٤‬خ ‪٘٣‬بظ ص‪ٓٞ‬بد اُم‪ٝ‬اٍ اُز‪ٝ‬ع‪٤‬خ اُزوِ‪٤‬م‪٣‬خ ‪ٌُٖ ،‬‬ ‫‪ ،‬ا‪ ١‬اٗ‪ ٚ‬ال‪ٗ ١‬وطخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ثط ‪٣‬وخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓغ االخا ثؼ‪ ٖ٤‬االػزجب أُؾ‪ٞ‬‬ ‫ٓؾز‪ٞ‬ا‪ ٙ‬ك‪ ٢‬اُغبٗت اال‪٣‬ض‬ ‫‪ٞ٣‬عم ٗوطخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ‬ ‫ٓؾز‪ٞ‬ا‪ ٙ‬ك‪ ٢‬اُغبٗت اال‪ُِٔ ٖٔ٣‬ؾ‪ٞ‬‬ ‫‪ٝ‬اُز‪ ٢‬ثم‪ٛ ٝ‬ب ر٘بظ اُ٘وطخ ‪ٝ ،‬ثرٌَ ٓزجب ٍ‪.‬‬ ‫ُِٔؾ‪ٞ‬‬ ‫ػ٘مٓب ٗضزاً صْ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ٗغم اٗ‪ٜ‬ب ٓصبؿخ ٖٓ ٗوبغ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ػِٔب‬ ‫إ اُ٘وبغ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ال رِٔي كوػ ثؼما صل ‪٣‬ب ثَ ا‪٣‬عب اثؼب ‪ٝ 1,2 ٢ٛ‬اًض ثبالػزٔب ػِ‪٠‬‬ ‫اُلعبءاد أُؼ كخ ػِ‪ٜ٤‬ب اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٝ‬اُز‪ ٢‬ر خا ٓ٘‪ٜ‬ب ه‪ٜٔ٤‬ب ًٔب ‪ٝ‬رؼزٔم ػِ‪ ٠‬ص‪٤‬ـخ‬ ‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ذار‪ٜ‬ب ‪.‬‬

‫‪ -15.2‬انذانح انُ​ُىذروصىفكُح انفردَح‬ ‫‪ُٜٝ‬ب اُزؼ ‪٣‬ق ٗلض‪ٚ‬‬ ‫ثط ‪٣‬وخ ٓربث‪ٜ‬خ ‪ٗ ،‬ضزط‪٤‬غ رؼ ‪٣‬ق اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُل ‪٣‬خ‬ ‫‪ٝ‬ذُي ُغٔ‪٤‬غ ه‪ ْ٤‬ك‪ ، ٢‬اال اٗ٘ب ‪٣‬غت إ‬ ‫ُِماُخ اُل ‪٣‬خ اُزوِ‪٤‬م‪٣‬خ ا‪ ١‬إ‬ ‫ؽ‪٤‬ش رٔضَ ٓ ًجخ اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م)‪.‬‬ ‫ٗظ‪٤‬ق اُزؼٔ‪ ْ٤‬اُزبُ‪٢‬‬ ‫كٔضال‬ ‫{‬

‫)‪(61‬‬

‫ك‪ ٢‬اُؾو‪٤‬وخ اُغزء اال‪ ٖٓ ٍٝ‬اُماُخ رْ ررٌ‪ ِٚ٤‬ث‪ٞ‬ظغ اُز‪ٓ ٖ٤‬خزِلز‪ٓ ٖ٤‬ؼب‪ ،‬ثطج‪٤‬ؼخ اُؾبٍ ‪ ٖٓٝ‬اعَ‬ ‫ٗالؽع إ‬ ‫‪, and‬‬ ‫–‬ ‫ٗغم ‪:‬‬ ‫ث‪ٔ٘٤‬ب ٖٓ اعَ‬ ‫;‬ ‫اذٕ ‪،‬‬

‫‪،‬ثبُزبُ‪ ٢‬كبٕ‬

‫رٔضَ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ك ‪٣‬خ‪.‬‬

‫‪35‬‬

‫‪Neutrosophic Science International Association‬‬

‫اُ صْ اُج‪٤‬بٗ‪)11( ٢‬‬ ‫‪ٝ‬ثرٌَ ٓربث‪ : ٚ‬اُماُخ اُل ‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓز٘بظ ح ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬ب ٗضجخ ُ٘وطخ األصَ ك‪٢‬‬ ‫اُ٘ظبّ االؽماص‪ ٢‬اٌُب ر‪٤‬ز‪.١‬‬

‫‪ - 16.2‬انًُىرض انُ​ُىذروصىفكٍ‬ ‫إ و‪ٞٔٗ ١‬ذط ‪ِٔ٣‬ي ش‪٢‬ء ٖٓ اُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ‪٣‬ضٔ‪ٞٔٗ ٠‬ذعب ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬ب‪ .‬ػ٘م عٔغ ث‪٤‬بٗبد‬ ‫رصق ػبُٔب ك‪٤‬ز‪٣‬بئ‪٤‬ب ٓؼ‪٘٤‬ب ‪ٝ‬رٌ‪ٛ ٕٞ‬ا‪ ٙ‬اُج‪٤‬بٗبد ؿ‪ ٤‬ربٓخ ‪ ،‬و‪ ٝ‬ك‪ٜ٤‬ب ش‪٢‬ء ٓج‪ ، ْٜ‬و‪ ٝ‬هم رؾز‪١ٞ‬‬ ‫ثؼط اُز٘بهعبد و ‪ ٝ‬ثٔب رٌ‪ ٕٞ‬ؿ‪ٝ ٤‬اظؾخ صٌ٘‪ ٕٞ‬ػ٘م‪ٛ‬ب ؿ‪ ٤‬هب ‪ ٖ٣‬ػِ‪ ٠‬ث٘بء ٗٔ‪ٞ‬ذط روِ‪٤‬م‪١‬‬ ‫ه‪٤‬ن‪ٛ ،‬اا ص‪٤‬و‪ٗ ٞ‬ب لُ‪ ٠‬ث٘بء ٗٔ‪ٞ‬ذط رو ‪٣‬ج‪( ٢‬ذ‪ ٝ‬صٔي)‪.‬‬ ‫كجبصزخماّ االؽصبء اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬ص٘ صْ اُج‪٤‬بٗبد صْ ٗصْٔ غ ‪٣‬وخ ُالٗؾما اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫‪ٝ‬لٕ اًض ‪ٛ‬ا‪ ٙ‬اُط م ش‪ ٜ‬ح ‪ ٢ٛ‬االٗؾما اُخط‪ ٢‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٝ ٢‬غ ‪٣‬وخ لٗؾما أُ ثؼبد‬ ‫اُصـ ‪ ٟ‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٖٓ وعَ ٓزـ‪ٞ٤ٗ ٖ٣ ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٓ ، ٝ ٖ٤٤‬زٔضِخ ثبُج‪٤‬بٗبد أُ ص‪ٓٞ‬خ‬ ‫‪ ٌٖٔ٣ ،‬أل‪ ١‬شخا ًبٕ رصٔ‪ ْ٤‬اكعَ ٓالءٓخ ٌٓٔ٘خ ُِٔ٘ؾ٘‪ ٢‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬ثط ‪٣‬وخ اٗؾما‬ ‫ٓؼ‪٘٤‬خ ثمال ٖٓ اُج‪٤‬بٗبد اُ‪ٜ‬رخ ‪ًٔ ،‬ب ٗغم‪ٛ‬ب ك‪ ٢‬االٗؾما اُزوِ‪٤‬م‪ٓ ، ١‬ضبٍ ذُي‬ ‫‪},‬‬

‫)‪(62‬‬

‫{‬

‫ص٘ؼَٔ ٓغ ٓغٔ‪ٞ‬ػخ ث‪٤‬بٗبد رو ‪٣‬ج‪٤‬خ ك‪ ٢‬االٗؾما اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪: ٢‬‬ ‫)‬ ‫)‪(63‬‬

‫}‬

‫)‬

‫(‬

‫{∈‬

‫كٔضال ثمال ٖٓ اُؾص‪ ٍٞ‬ػِ‪ ٠‬اٗؾما خط‪ٛ ٢‬ش ًٔب ك‪ ٢‬ػِْ االؽصبء اُزوِ‪٤‬م‪: ١‬‬ ‫‪36‬‬

‫‪Neutrosophic Science International Association‬‬ ‫)‪(64‬‬ ‫ص٘ؾصَ ػِ‪ٓ ٠‬غٔ‪ٞ‬ػخ اٗؾما خط‪ًٔ ٢‬ب ك‪ ٢‬أُضبٍ اُزبُ‪.٢‬‬ ‫)‪(65‬‬ ‫‪ٛ‬اا ٓب ٗغم‪ ٙ‬ك‪ ٢‬ػِْ االؽصبء اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪.٢‬‬

‫‪ - 12.2‬يؼايم االرذثاط انُ​ُىذروصىفكٍ‬ ‫ٓوب ٗخ ثٔؼبَٓ اال رجبغ اُزوِ‪٤‬م‪ٝ ١‬اُا‪ٗ ١‬غم‪ ٙ‬ػجب ح ػٖ ػم ‪ٛ‬ش ‪٘٣‬زٔ‪ ٢‬اُ‪ ٠‬اُلز ح‬ ‫]‪ٗ [-1, 1‬غم إ ٓؼبَٓ اال رجبغ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬ػجب ح ػٖ ٓغٔ‪ٞ‬ػخ عزئ‪٤‬خ ٖٓ اُلز ح ]‪.[-1, 1‬‬ ‫‪ٝ‬ثط ‪٣‬وخ ٓربث‪ٜ‬خ ‪ ،‬لذا ًبٕ وؿِت ‪ٛ‬ا‪ ٙ‬أُغٔ‪ٞ‬ػخ اُغزئ‪٤‬خ ُٔؼبَٓ اال رجبغ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٝ ٢‬ا رجبغ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ ٢‬ا‪٣‬غبث‪ٓٝ ، ٢‬ب ػما ذُي ص‪ُ ٌٕٞ٤‬م‪ٜ٣‬ب ا رجبغ‬ ‫ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬صِج‪. ٢‬‬ ‫ثطج‪٤‬ؼخ اُؾبٍ ‪ ،‬ال ٗغم ٗٔ‪ٞ‬ذعب ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬ب ‪ٝ‬ؽ‪٤‬ما ُٔربًَ اُؾ‪٤‬بح اُ‪ٞ‬اهؼ‪٤‬خ ‪ٝ ،‬ثبُزبُ‪ ٢‬ال ر‪ٞ‬عم‬ ‫ه‪ٞ‬اٗ‪ٞ٤ٗ ٖ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓعج‪ٞ‬غخ الصزخمآ‪ٜ‬ب ك‪ ٢‬اُ٘ٔبذط اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪.‬‬ ‫إ ًَ ٗٔ‪ٞ‬ذط ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٣ ٢‬ؼم ٗٔ‪ٞ‬ذعب رو ‪٣‬ج‪٤‬ب ًٔب ‪ٝ‬إ اُزو ‪٣‬جبد ثٔب رٌ‪ٖٓ ٕٞ‬‬ ‫‪ٝ‬ع‪ٜ‬بد ٗظ ٓخزِلخ ‪ .‬اُ٘ٔ‪ٞ‬ذط هم ‪٣‬ؼزج اكعَ ٖٓ ؿ‪ ٙ ٤‬اذا ًبٗذ رخٔ‪٘٤‬بر‪ ٚ‬اكعَ ٖٓ رخٔ‪٘٤‬بد‬ ‫اُ٘ٔبذط االخ ‪ ٌُٖ ٟ‬ك‪ ٢‬اؿِت اُؾبالد‪ ،‬اُ٘ٔ‪ٞ‬ذط هم ‪ ٌٕٞ٣‬وكعَ ٖٓ ‪ٝ‬ع‪ٜ‬خ ٗظ ‪ٝ ،‬اص‪ٞ‬و ٖٓ‬ ‫‪ٝ‬ع‪ٜ‬خ ٗظ اخ ‪ ٟ‬ذُي الٕ ٓربًَ اُؾ‪٤‬بح اُ‪ٞ‬اهؼ‪٤‬خ رؼزٔم ‪ٝ‬ثرٌَ غج‪٤‬ؼ‪ ٢‬ػِ‪ ٠‬اُؼم‪٣‬م ٖٓ أُؼبٓالد‬ ‫ٓ٘‪ٜ‬ب ٓب ‪ٓ ٞٛ‬ؼ ‪ٝ‬ف ‪ٜ٘ٓٝ‬ب ٓب ‪ ٞٛ‬ؿ‪ٓ ٤‬ؼ ‪ٝ‬ف‪.‬‬ ‫‪ٓٝ‬غ ذُي ‪٘ٛ ،‬بُي ؽبعخ ُِ٘ٔبذط اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اُز‪ ٢‬رٔضَ اُ‪ٞ‬اهغ ‪ ٖٓٝ‬وعَ رض ‪٣‬غ رؾِ‪ َ٤‬اُجمائَ‬ ‫(أُززبثؼبد) ًاُي ال‪٣‬غب ؽِ‪ٓ ٍٞ‬ضِ‪ ٠‬رو ‪٣‬ج‪٤‬خ‪.‬‬

‫‪ -12.2‬انذانح االصُح انُ​ُىذروصىفكُح‬ ‫اُماُخ األص‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٢ٛ‬اُخ وص‪٤‬خ رِٔي ثؼط اُالرؼ‪ ، ٖ٤٤‬إ ‪ٛ‬اا اُالرؼ‪ ٖ٤٤‬هم‬ ‫‪ ٌٕٞ٣‬ك‪ ٢‬لؽم‪ ٟ‬ا‪ ٝ‬وًض ٓٔب ‪: ٢ِ٣‬‬ ‫‪ -1‬ك‪ ٢‬ص‪٤‬ـخ اُماُخ ( ص‪ٞ‬اء ك‪ ٢‬االس ا‪ ٝ‬االصبس )‪.‬‬ ‫‪ -1‬ك‪٘ٓ ٢‬طِن اُماُخ ‪.‬‬ ‫‪ -1‬ك‪ٓ ٢‬م‪ ٟ‬اُماُخ ‪.‬‬ ‫ر َٓ اُماُخ األص‪٤‬خ اُزوِ‪٤‬م‪٣‬خ‪:‬‬ ‫)‪(66‬‬ ‫ؽ‪٤‬ش وٕ‬

‫‪ٝ‬‬

‫‪,‬‬ ‫كِ‪ً ٞ‬بٕ اُالرؼ‪ ٖ٤٤‬ك‪ ٢‬اصبس اُماُخ ‪ٓ ،‬ضبٍ ذُي اٗظ اُماُخ اُزبُ‪٤‬خ‪:‬‬ ‫‪37‬‬

‫‪Neutrosophic Science International Association‬‬ ‫‪,‬‬ ‫)‪(67‬‬ ‫ػجب ح ػٖ كز ح رؾز‪ ١ٞ‬اُ هْ ‪ ،1‬كض٘ؼَٔ ثاُي ػِ‪ ٠‬اُماُخ ذاد اُضٔي ًٔب ‪:٢ِ٣‬‬ ‫ؽ‪٤‬ش إ‬

‫اُ صْ اُج‪٤‬بٗ‪)14( ٢‬‬ ‫‪ً ُٞٝ‬بٕ اُالرؼ‪ٞٓ ٖ٤٤‬ع‪ ٞ‬ا ك‪ ٢‬األس‪ ،‬الؽع اُماُخ‬ ‫)‪(68‬‬

‫اُ صْ اُج‪٤‬بٗ‪)11( ٢‬‬ ‫ٖٓ أُ‪ ْٜ‬االٗزجب‪ ٙ‬اُ‪٤ً ٠‬ل‪٤‬خ صِ‪ٞ‬ى اُماُخ‬

‫كٔضال‬

‫)‪(69‬‬ ‫‪38‬‬

‫‪Neutrosophic Science International Association‬‬ ‫و‪ ١‬اٗ٘ب ُض٘ب ٓز ًم‪ َٛ ٖ٣‬ص‪ ٞ‬ح‬

‫‪ 2 ٢ٛ‬اّ ‪. 4‬‬

‫ٓضبٍ اخ ُماُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ اص‪٤‬خ‬ ‫)‪(70‬‬ ‫ؽ‪٤‬ش االصش ك‪٢‬‬ ‫إ ‪ٛ‬ا‪ ٙ‬اُماُخ رخزِق اخزالكب عا ‪٣‬ب ػٖ اُماُخ‬ ‫ذاد صٔي ًٔب ٓالؽع ك‪ ٢‬اُرٌَ اُزبُ‪٢‬‬ ‫وٕ اُماُخ‬

‫رٔضَ كز ح ُاا ٗز‪ٞ‬هغ‬

‫اُ صْ اُج‪٤‬بٗ‪)16( ٢‬‬ ‫‪٘​ٌ٘ٔ٣‬ب اصز٘زبط وٕ ُِماُخ‬

‫صٔي ٖٓ اُزؼ‪٣ٞ‬ط اُجض‪٤‬ػ اُزبُ‪٢‬‬

‫)‪(71‬‬ ‫‪ ٢ٛٝ‬كز ح ٓلز‪ٞ‬ؽخ‪.‬‬

‫‪ -12.2‬انذانح انهىغارَرًُح انُ​ُىذروصىفكُح‬ ‫ػِ‪ ٠‬ؿ ا ٓب الؽظ٘ب‪ ٖٓ ٙ‬اُج٘م اُضبثن ُِماُخ األص‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٗ ،‬غم وٕ اُماُخ اُِ‪ٞ‬ؿب ‪٣‬زٔ‪٤‬خ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٢ٛ‬و‪٣‬عب اُخ ُ‪ٞ‬ؿب ‪٣‬زٔ‪٤‬خ رِٔي ثؼط اُالرؼ‪ ، ٖ٤٤‬لذ وٕ ‪ٛ‬اا اُالرؼ‪ ٖ٤٤‬هم ‪ٌٕٞ٣‬‬ ‫ك‪ ٢‬اؽم‪ ٟ‬ا‪ ٝ‬اًض االٓبًٖ األر‪٤‬خ ‪:‬‬ ‫‪ -1‬ك‪ ٢‬ص‪٤‬ـخ اُماُخ‪.‬‬ ‫‪ -1‬ك‪ٓ ٢‬غبٍ اُماُخ ‪.‬‬ ‫‪ -1‬ك‪ٓ ٢‬غبُ‪ٜ‬ب أُوبثَ ‪.‬‬ ‫ٓضال‬ ‫)‪(72‬‬

‫‪39‬‬

Neutrosophic Science International Association

)11( ٢ٗ‫ب‬٤‫اُ صْ اُج‬ ‫ٓضبٍ آخ‬ (

)

(73)

)12( ٢ٗ‫ب‬٤‫اُ صْ اُج‬ ‫ ا ٗالؽع‬٤‫اخ‬ (74) 41

‫‪Neutrosophic Science International Association‬‬

‫اُ صْ اُج‪٤‬بٗ‪)13( ٢‬‬

‫‪ - 21.2‬ذركُة انذوال انُ​ُىذروصىفكُح‬ ‫ثرٌَ ػبّ ػ٘م ر ً‪٤‬ت اُز‪ٞ٤ٗ ٖ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬ز‪ ٖ٤‬كبٕ اُالرؼ‪ ٖ٤٤‬ص‪٤‬ز ا ‪ٓٝ .‬ضبٍ ذُي‬

‫‪ٝ‬ثبُزبُ‪ ٢‬كنٕ‬ ‫)‪(75‬‬

‫)‬

‫(‬

‫‪40‬‬

‫‪Neutrosophic Science International Association‬‬

‫انفصم انصانس‬ ‫حضاب انرفاضم وانركايم انُ​ُىذروصىفٍ‬ ‫‪ -1.3‬انغاَح انُ​ُىذروصىفكُح‬ ‫‪٣‬وصم ثبُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ؿب‪٣‬خ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪،‬لذ هبّ ٓؤصش أُ٘طن‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬ثز‪ٞ‬ص‪٤‬غ أُل‪ ّٜٞ‬اُزوِ‪٤‬م‪ُِ ١‬ـب‪٣‬خ ُ٘ خا ث٘ظ االػزجب اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪:‬‬

‫ٓزٔضِخ ثبُ صْ اُج‪٤‬بٗ‪ ٢‬االر‪:٢‬‬

‫اُ صْ اُج‪٤‬بٗ‪)12( ٢‬‬ ‫{‬

‫)‪(76‬‬ ‫‪ ٢ٛ‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ذاد عزئ‪. ٖ٤‬‬ ‫‪ٛٝ‬اا ‪ٞ٣‬ظؼ إ‬ ‫ثبصزخماّ غ ‪٣‬وخ اُ صْ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ص٘ؾصَ ػِ‪:٠‬‬ ‫‪ ‬اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٖٓ اُ‪٤‬ضب ‪:٢ٛ‬‬ ‫;‬ ‫)‪(77‬‬ ‫‪‬‬ ‫)‪(78‬‬

‫اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٖٓ اُ‪:٢ٛ ٖ٤ٔ٤‬‬ ‫‪.‬‬

‫‪42‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ص‪٤‬ومّ أُؤُق ك‪ٛ ٢‬اا اُج٘م ‪ٝ‬أل‪ ٓ ٍٝ‬ح ٓل‪ ّٜٞ‬شج‪ ٚ‬اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪ ،‬لذ ‪٣‬غت إ ٗؼِْ‬ ‫ع‪٤‬ما إ شج‪ ٚ‬اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٢ٛ‬روبغغ ُِـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٖٓ اُ‪٤‬ضب ٓغ اُـب‪٣‬خ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٖٓ اُ‪ٓ ] ٖ٤ٔ٤‬ربث‪ٜ‬ب ُٔب ٗغم‪ ٙ‬ك‪ٞٓ ٢‬ظ‪ٞ‬ع اُـب‪٣‬بد اُزوِ‪٤‬م‪ ، ١‬لذ ٗغم وٕ اُـب‪٣‬خ ٖٓ‬ ‫اُ‪٤‬ضب ‪٣‬غت ُ‪ٜ‬ب إ رضب‪ ١ٝ‬اُـب‪٣‬خ ٖٓ اُ‪ٛٝ ،ٖ٤ٔ٤‬اا ك‪ ٢‬اُؾو‪٤‬وخ ‪ٌ٣‬بكئ اُزوبغغ ث‪ ٖ٤‬اُـب‪٣‬خ ٖٓ اُ‪٤‬ضب‬ ‫‪ ،‬ا‪ٝ‬‬ ‫(‪ٗٝ‬وصم ثاُي أُغٔ‪ٞ‬ػخ أٌُ‪ٗٞ‬خ ٖٓ هْ ٓ٘ز‪ٓ ٢ٜ‬ل ا‪ ٖٓ ٝ‬و هبّ ؿ‪٘ٓ ٤‬ز‪٤ٜ‬خ ثبرغب‪ٙ‬‬ ‫) ‪ٝ‬ث‪ ٖ٤‬اُـب‪٣‬خ ٖٓ اُ‪ٗ ( ٖ٤ٔ٤‬وصم ثاُي أُغٔ‪ٞ‬ػخ أٌُ‪ٗٞ‬خ ٖٓ هْ‬ ‫ا هبّ ؿ‪٘ٓ ٤‬ز‪٤ٜ‬خ ثبرغب‪ٙ‬‬ ‫) لذ وٕ ‪ٛ‬اا‬ ‫‪ ،‬ا‪ ٝ‬ا هبّ ؿ‪٘ٓ ٤‬ز‪٤ٜ‬خ ثبرغب‪ٙ‬‬ ‫ٓ٘ز‪ٓ ٢ٜ‬ل ‪ ،‬ا‪ ٝ‬ا هبّ ؿ‪٘ٓ ٤‬ز‪٤ٜ‬خ ثبرغب‪ٙ‬‬ ‫اُزوبغغ ال ‪ ٌٕٞ٣‬خبُ‪٤‬ب[‬ ‫ًٔب ٗغم ذُي ‪ٝ‬اظؾب ك‪ٔ٤‬ب ‪:٢ِ٣‬‬ ‫)‪(79‬‬ ‫اذا ًبٕ اُزوبغغ ث‪ ٖ٤‬اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٖٓ اُ‪٤‬ضب ‪ٝ‬اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٖٓ اُ‪ٖ٤ٔ٤‬‬ ‫‪ٔ٣‬ضَ ٓغٔ‪ٞ‬ػخ خبُ‪٤‬خ ػ٘مئا شج‪ ٚ‬اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ؿ‪ٞٓ ٤‬ع‪. ٞ‬‬ ‫ٓ‪ٞ‬ع‪ ٞ‬ح اذا ًبٗزب اُـب‪٣‬زبٕ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬زبٕ ٖٓ‬ ‫إ اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ُِماُخ‬ ‫اُ‪٤‬ضب ‪ ٖٓٝ‬اُ‪ٓ ٖ٤ٔ٤‬زطبثوز‪ ٓٝ ( ،ٖ٤‬ح وخ ‪٣ ٗ ٟ‬م إ ٗاً ث ٕ اُـب‪٣‬ز‪ ٖ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬ز‪ٖٓ ٖ٤‬‬ ‫اُ‪٤‬ضب ‪ ٖٓٝ‬اُ‪ٔٛ ٖ٤ٔ٤‬ب ػجب ح ػٖ ٓغبٓ‪٤‬غ ‪ ،‬ثمال ٖٓ االػما )‬ ‫‪.‬‬ ‫كٔضال ‪ :‬اُماُخ اُضبثوخ الرِٔي ؿب‪٣‬خ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ؛ ألٕ‬

‫انًؼُار‬ ‫ص٘ؼ ف أُؼ‪٤‬ب ًٔب ‪. ٢ِ٣‬‬ ‫ُزٌٖ‬ ‫(‪)80‬‬ ‫‪ٓ ٢ٛ‬غٔ‪ٞ‬ػخ اُو‪ ِ ٟٞ‬ث‪ٔ٘٤‬ب رٔضَ ٓغٔ‪ٞ‬ػخ ًَ االػما اُؾو‪٤‬و‪٤‬خ ‪ .‬أل‪١‬‬ ‫ؽ‪٤‬ش إ‬ ‫∈‬ ‫ٓغٔ‪ٞ‬ػخ‬ ‫∈ }| |{‬ ‫‪},‬‬ ‫)‪(81‬‬ ‫)‪ (frontier‬و‪ ١‬وٕ ‪:‬‬ ‫‪ ٢ٛ‬ؽم‪ٝ‬‬ ‫ٗالؽع وٕ | | رٔضَ اُو‪ٔ٤‬خ أُطِوخ ِ ‪ٝ‬‬ ‫|{‬ ‫||‬ ‫}|‬ ‫)‪(82‬‬ ‫رؼ٘‪ ٢‬وصـ ه‪ٔ٤‬خ ػظٔ‪( ٠‬وصـ‬ ‫رؼ٘‪ ٢‬وًج ه‪ٔ٤‬خ صـ ‪ ٟ‬ك‪ٝ ، ٢‬‬ ‫لذ إ‬ ‫ٗ‪ٜ‬ب‪٣‬خ ػظٔ‪ )٠‬ك‪ . ٢‬ػ٘مئا ٗغم‬ ‫|{‬ ‫||‬ ‫}|‬ ‫| | |{‬ ‫| | ||‬ ‫}|‬ ‫)‪(83‬‬ ‫ؽ‪٤‬ش ∈ ‪ٔ٣‬ضَ ه‪٤‬بص‪( ٢‬ؿ‪ٓ ٤‬زغ‪ )ٚ‬لذا ًبٗذ أُغٔ‪ٞ‬ػخ ٌٓ‪ٗٞ‬خ ٖٓ ػ٘ص ‪ٝ‬اؽم كوػ ‪ ،‬و‪١‬‬ ‫‪| |.‬‬ ‫)‪(84‬‬ ‫‪ٔ٣‬ضَ ٓؼ‪٤‬ب ا ِ‬ ‫ص٘ج ‪ ٖٛ‬إ‬ ‫∈ | |{‬ ‫}‬ ‫∈‬ ‫|{‬ ‫||‬ ‫}|‬ ‫)‪(85‬‬ ‫| | |{‬ ‫| | ||‬ ‫}|‬ ‫|{‬ ‫||‬ ‫}|‬ ‫)‪(86‬‬ ‫‪43‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ٖٓ وعَ اُو‪٤‬بص‪. ٢‬‬ ‫||‬ ‫}|‬ ‫)‪(87‬‬ ‫|{‬ ‫|‬

‫|{‬

‫||‬

‫| |||‬

‫}|‬

‫||‬ ‫||‬

‫}|‬

‫}|‬ ‫)‪(88‬‬ ‫)‪(89‬‬

‫| | |{‬

‫}|‬

‫||‬

‫|{‬

‫|{‬ ‫|‬ ‫}|‬

‫|‬ ‫||‬

‫| |‬ ‫|{‬

‫|‬

‫‪ -2.3‬يالءيح انثؼذ‪ -‬انجزئٍ (انًررٌ – انجزئٍ)‬ ‫ًِ‪ٜ‬ب ػجب ح‬ ‫‪ٓ ٝ‬غٔ‪ٞ‬ػزبٕ ك‪ ، ٢‬ثؾ‪٤‬ش إ‬ ‫ُزٌٖ‬ ‫‪٣ ٝ‬ؼ ف ثبُرٌَ اُزبُ‪:٢‬‬ ‫ػٖ وػما ٓ٘ز‪٤ٜ‬خ‪ .‬إ اُجؼم‪ -‬اُغزئ‪( ٢‬أُز ‪ – ١‬اُغزئ‪ )٢‬ث‪ٖ٤‬‬ ‫‪2‬‬ ‫‪+‬‬ ‫‪η:‬‬ ‫)‪η(A, B) = max{|infA-infB|, |supA-supB|}.‬‬ ‫)‪(90‬‬ ‫ثؼجب ح اخ ‪ ، ٟ‬إ ٓالءٓخ اُجؼم‪ -‬اُغزئ‪( ٢‬أُز ‪ – ١‬اُغزئ‪٣ )٢‬و‪٤‬ش ً‪٤‬ل‪٤‬خ اؿالم اًج اُو‪ْ٤‬‬ ‫اُصـ ‪ٝ ، ٟ‬اؿالم اصـ اُو‪ ْ٤‬اُؼظٔ‪ُٔ sup’s ٝ inf’s ٠‬غٔ‪ٞ‬ػز‪ٗ( ٖ٤‬وصم ثاُي ‪،‬‬ ‫أُغٔ‪ٞ‬ػزبٕ ثبُ٘ظ ُؾم‪ٔٛ ٝ‬ب اُوص‪)ٟٞ‬‬

‫‪ -3.3‬خىاص يالءيح انثؼذ‪ -‬انجزئٍ (انًررٌ – انجزئٍ)‬ ‫لذ إ‬ ‫ال‪ ١‬صالس ٓغٔ‪ٞ‬ػبد ‪ٓ A, B, C‬ؾز‪ٞ‬اح ك‪٢‬‬ ‫ًِ‪ٜ‬ب رِٔي اػما ٓ٘ز‪٤ٜ‬خ ‪ ،‬ث‪ٞ‬صؼ٘ب اُو‪ ٍٞ‬وٕ ‪:‬‬ ‫‪a) η(A, B) ≥ 0.‬‬ ‫)‪(91‬‬ ‫‪b) η(A, A) = 0.‬‬ ‫)‪(92‬‬ ‫ُ‪ً ٞ‬بٗذ ‪ٛ η(A, B) = 0‬اا ال‪٣‬ؼ٘‪ ٢‬ثبُع ‪ ٝ‬ح إ ‪ A ≡ B‬ثَ إ ذُي ‪٣‬ؼ٘‪ُ٘ ٢‬ب إ‬ ‫‪.supA = supBٝ ، infA = infB‬‬ ‫ٓضال اذا ًبٗذ ‪ ، A = {3, 4, 5, 7} and B = (3, 7],‬ػ٘مئا ‪infA = infB = 3 and‬‬ ‫‪ُ.‬اُي كبٕ ٓضِٔخ اُجؼم ‪ٛ‬ا‪ٙ‬‬ ‫ٌُٖ‬ ‫‪ supA = supB = 7‬ثؾ‪٤‬ش إ‬ ‫رزؾون كوػ ثرٌَ عزئ‪ ٢‬ث‪ٞ‬اصطخ ‪.‬‬ ‫‪c) η(A, B) = η(B, A).‬‬ ‫)‪(94‬‬ ‫‪d) η(A, B) ≤ η(B, C)+ η(C, A).‬‬ ‫)‪(95‬‬ ‫‪Proof of d):‬‬ ‫|{‬ ‫||‬ ‫}|‬ ‫|{‬ ‫||‬ ‫}|‬ ‫)‪(96‬‬ ‫‪But‬‬ ‫|‬ ‫|‬ ‫|‬ ‫|‬ ‫|‬ ‫|‬ ‫|‬ ‫|‬ ‫|‬ ‫|‬ ‫)‪(97‬‬ ‫‪ٝ‬ثرٌَ ٓربث‪ٚ‬‬ ‫‪44‬‬

‫‪Neutrosophic Science International Association‬‬ ‫|‬

‫|‬

‫|‬

‫|‬

‫|‬ ‫|‬

‫)‪(98‬‬

‫|‬

‫|‬

‫|‬

‫‪ٝ‬ثبُزبُ‪ ٢‬كبٕ‪:‬‬ ‫||‬ ‫|{‬

‫}|‬

‫|{‬

‫||‬ ‫}|‬

‫)‪(99‬‬

‫∈ ‪ٗ a, b‬وصم ثاُي إ‬

‫|{‬ ‫}|‬ ‫||‬ ‫‪ٝ‬‬

‫‪٣‬ؾز‪٣ٞ‬بٕ‬

‫‪ )e‬اذا ًبٗذ }‪ٓ B = {b} ٝ A = {a‬غ اُؼِْ إ‬ ‫ًَ ٓ٘‪ٜٔ‬ب ػِ‪ ٠‬ػ٘ص ‪ٝ‬اؽم كوػ ػ٘مئا‪.‬‬ ‫|‬ ‫|‬ ‫)‪(100‬‬ ‫‪ ) f‬اذا ًبٗذ ‪ً ٝ‬ال‪ٔٛ‬ب كز اد (ٓلز‪ٞ‬ؽخ ‪ٓ ،‬ـِوخ‪،‬ا‪ ٝ‬شج‪ٓ ٚ‬لز‪ٞ‬ؽخ ‪ /‬شج‪ٓ ٚ‬ـِوخ)‪.‬‬ ‫ٓضال ]‪ٓ B = [b1, b2 ] ٝ A = [a1, a2‬غ االخا ثؼ‪ ٖ٤‬االػزجب إ ‪ b1 < b2, ٝ a1 < a2‬ػ٘مئا‬ ‫|{‬ ‫||‬ ‫}|‬ ‫)‪(101‬‬

‫‪ -4.3‬انفضاء انًررٌ – انجزئٍ‬ ‫رٔضَ‬

‫ٓغٔ‪ٞ‬ػخ ؿ‪ ٤‬خبُ‪٤‬خ ‪ ،‬اُماُخ‬ ‫ؽ‪٤‬ش‬ ‫ثرٌَ ػبّ ٗل ض وٕ ُم‪٘٣‬ب اُماُخ‬ ‫‪ٝ ،‬رؼ ف ًٔب ‪:٢ِ٣‬‬ ‫أُز ‪ - ١‬اُغزئ‪ ( ٢‬اُجؼم – اُغزئ‪ )٢‬ػِ‪٠‬‬ ‫|{‬ ‫||‬ ‫}|‬ ‫)‪(102‬‬ ‫ٓزٔزغ ثماُخ ٓضَ ‪٣‬ضٔ‪ ٠‬اُلعبء أُز ‪ – ١‬اُغزئ‪ٛ . ٢‬اا أُز ‪ ١‬اُغزئ‪ِ ٢‬‬ ‫لٕ اُلعبء‬ ‫لذ ‪ ٢ٛ‬و‪ٓ ١‬غٔ‪ٞ‬ػخ‬ ‫‪ٔ٣‬ضَ رؼٔ‪ٔ٤‬ب ُِٔز ‪ d ١‬أُؼ ف ك‪ ٢‬رؾِ‪ َ٤‬اُلز ح ‪ًٔٝ‬ب ‪٢ِ٣‬‬ ‫ؽو‪٤‬و‪٤‬خ ‪ٝ‬‬ ‫|{‬ ‫||‬ ‫‪|},‬‬ ‫)‪(103‬‬ ‫‪ٝ ،‬ألٕ ‪ٌٜ٘ٔ٣‬ب وٕ رزؼبَٓ ٓغ ًَ وٗ‪ٞ‬اع أُغبٓ‪٤‬غ‪٤ُٝ ،‬ش كوػ‬ ‫‪ٝ‬‬ ‫ػِٔب وٕ‬ ‫اُلز اد ًٔب ك‪ ٢‬اُزؾِ‪ َ٤‬اُصؾ‪٤‬ؼ‪.‬‬ ‫‪ ٖٓٝ‬اُالكذ ُِ٘ظ وٕ ‪:‬‬ ‫|{‬ ‫|‬ ‫|‬ ‫}|‬ ‫|{‬ ‫||‬ ‫}|‬ ‫)‪(104‬‬ ‫‪ٛٝ‬اا ‪ٔ٣‬ضَ ثبُعجػ ٓؼ‪٤‬ب ‪.‬‬

‫‪ - 5.3‬ذؼرَف‬

‫نهغاَح انُ​ُىذروصىفكُح انُضري‬

‫ُِـب‪٣‬خ اُ‪٤‬ض ‪ٟ‬‬ ‫‪ ،‬لٕ رؼ ‪٣‬ق‬ ‫ُزٌٖ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ،‬ؽ‪٤‬ش‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٞٛ‬ر‪ٞ‬ص‪٤‬غ ُِزؼ ‪٣‬ق اُزوِ‪٤‬م‪ُِ ١‬ـب‪٣‬خ ٖٓ اُ‪٤‬ضب ‪ ،‬لذ لٕ اُو‪ٔ٤‬خ أُطِوخ | | ‪٣‬زْ‬ ‫و‪٣‬عب‪ ،‬ثمال ٖٓ اُؼَٔ ٓغ ه‪ ْ٤‬ه‪٤‬بص‪٤‬خ كوػ ‪ ،‬ص٘ؼَٔ ا‪٣‬عب ٓغ ػ٘بص رٔضَ‬ ‫اصزجماُ‪ٜ‬ب ث )‬ ‫ٓغبٓ‪٤‬غ ( لذ إ أُغٔ‪ٞ‬ػخ رظ‪ًٔ ٜ‬ب ُ‪ ٞ‬اٗ‪ٜ‬ب رو ‪٣‬ت ُِو‪ٔ٤‬خ اُو‪٤‬بص‪٤‬خ)‪.‬‬ ‫‪45‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ثاُي كبٕ‬ ‫)‪(105‬‬ ‫‪ε‬‬

‫‪ٞ٣ ، ε‬عم‬

‫رٌبكئ ُِٔل‪ ّٜٞ‬اُ ‪٣‬بظ‪ ٢‬االر‪ٌَُ: ٢‬‬ ‫ػ٘مئا‬ ‫‪ε.‬‬ ‫)‪(106‬‬ ‫ُِـب‪٣‬خ اُ‪ ٠٘ٔ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ ٌٖٔ٣‬اُزؼج‪ ٤‬ػ٘‪ٜٔ‬ب ث‬ ‫لٕ رؼ ‪٣‬ق‬ ‫)‪(107‬‬ ‫‪ ٢ٛٝ‬رٌبكئ ٓب‪ٌَُ : ٢ِ٣‬‬ ‫كبٕ‬ ‫)‪(108‬‬ ‫‪ٝ‬ثرٌَ ػبّ كبٕ رؼ ‪٣‬ق‬ ‫‪ε‬‬ ‫‪ٞ٣ ε‬عم‬ ‫)‪(109‬‬

‫‪ٞ٣ ، ε‬عم‬

‫‪ε‬‬

‫لذ ُ‪ً ٞ‬بٕ‬

‫ثؾ‪٤‬ش اٗ‪ً ُٞ ٚ‬بٕ‬ ‫‪ε.‬‬ ‫‪ٌ٣‬بكئ اٗ‪ٌَُ ٚ‬‬

‫ُِـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ثؾ‪٤‬ش اٗ‪ً ُٞ ٚ‬بٗذ‬

‫ػ٘مئا‬ ‫‪ε.‬‬

‫‪ - 6.3‬يصال نحضاب انغاَح انُ​ُىذروصىفكُح‬ ‫ك‪ ٢‬أُضبٍ اُضبثن أُاً‪ ٞ‬ك‪ ٢‬صلؾخ ‪ً ُٞ ،14‬بٗذ‬ ‫|{‬ ‫}|‬ ‫‪ε‬‬

‫)‪(110‬‬ ‫ًٔب ك‪ ٢‬اُؾضجبٕ اُزوِ‪٤‬م‪ ١‬إ‬ ‫إ أُز اعؾخ ‪|} ε‬‬ ‫‪| ε‬‬ ‫)‪(111‬‬

‫| ‪ε ٝ‬‬

‫‪ُٝ‬زٌٖ‬

‫}|‬

‫|‬

‫| |‬ ‫|{‬

‫||‬ ‫|‬

‫رؼ٘‪٢‬‬ ‫||‬

‫‪ ε‬كبٕ‬

‫|{‬

‫|‪.‬‬ ‫رؼ٘‪ ٢‬إ أُز اعؾز‪ٖ٤‬‬

‫| رزؾووبٕ ػ٘مٓب‬ ‫‪.‬‬

‫‪and‬‬

‫|‬

‫|‬

‫‪ 2.3‬حانح خاصح نحضاب انغاَح انُ​ُىذروصىفكُح‬ ‫اك ض ( ًؾبُخ خبصخ ٖٓ أُضبٍ اُضبثن أُ‪ٞ‬ع‪ ٞ‬ك‪ ٢‬اُلو ح ‪ ) 1.1‬إ‬ ‫ٖٓ اُ‪ ٖ٤ٔ٤‬و‪ ٝ‬اُ‪٤‬ضب‬ ‫ًَ ٓ٘‪ٜ‬ب اُخ ثغزئ‪ ٖ٤‬لذ إ ع‪ٞ‬ا‬ ‫‪:ٞٛ‬‬ ‫∈ ‪, for‬‬ ‫;‬ ‫)‪(112‬‬ ‫∈ ‪, for‬‬ ‫;‬ ‫)‪(113‬‬ ‫∈ ‪, for‬‬ ‫;‬ ‫)‪(114‬‬ ‫‪46‬‬

‫‪Neutrosophic Science International Association‬‬ ‫)‪(115‬‬ ‫ُاُي كبٕ ‪،‬‬

‫∈‬

‫‪.‬‬ ‫|‬

‫|‬

‫‪, for‬‬ ‫|‬

‫|‬

‫‪ٝ‬ألٕ‬

‫∈‬

‫ػ٘مٓب‬

‫|‬

‫|‬

‫;‪ε‬‬

‫)‪(116‬‬

‫|‬ ‫|‬

‫‪ٛ ،‬اا ‪٣‬مػ‪ٗٞ‬ب ألٕ ٗ خا‬

‫ًاُي ُم‪٘٣‬ب‬ ‫|‬

‫|‬

‫|‬

‫|‬

‫|‬ ‫;‪ε‬‬

‫)‪(117‬‬ ‫‪٘ٛ‬ب اخا‬

‫|‬

‫‪ٝ‬ذُي ألٕ‬

‫‪ ,‬ػ٘مٓب‬

‫‪,‬‬

‫∈ ‪.‬‬ ‫ثاُي ص٘ؾصَ ػِ‪ ٠‬اٗ‪ٌَُ ٚ‬‬

‫‪ε‬‬

‫‪ε‬‬

‫‪-‬‬

‫‪ε ε‬‬

‫‪,‬‬

‫‪ٛٝ‬اا ٓب رٔضِ‪ ٚ‬ه‪ٔ٤‬خ اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ٖٓ اُ‪٤‬ضب ثط ‪٣‬وخ ٓربث‪ٜ‬خ ٗغم اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ٖٓ اُ‪ُٜ ٖ٤ٔ٤‬اا أُضبٍ‪.‬‬ ‫‪ٗ ε‬غم‬ ‫ُزٌٖ‬ ‫| |‬ ‫)‪(118‬‬

‫‪ε,‬‬

‫ٓٔب ‪٣‬ؼ٘‪ ٢‬وٗ‪ ٚ‬ػ٘مٓب‬

‫|‬

‫ُاُي ٗغم وٕ‬ ‫)‪(119‬‬ ‫ػِٔب وٕ ‪ε‬‬ ‫)‪(120‬‬

‫|{‬ ‫||‬

‫}|‬ ‫| ‪ٝ‬‬

‫}|‬

‫|{‬

‫ص٘ؾصَ ػِ‪٠‬‬ ‫‪| ε,‬‬ ‫| ‪ε, and‬‬ ‫|‬

‫‪ε‬‬

‫|‬

‫|‬

‫|‬ ‫|‬

‫|‬ ‫|‬

‫|‬

‫‪ٝ‬‬ ‫‪ε‬‬

‫|‬

‫|‬

‫|‬

‫|‬

‫|‬

‫|‬

‫ك‪ٓ ٢‬ؼب ُخ (‪ )120‬اػزج ٗب‬ ‫الؽظ٘ب اٗ‪ٌَُ ٚ‬‬ ‫)‪(121‬‬

‫‪ٞ٣ ε‬عم‬ ‫‪,‬‬

‫‪,ε -‬‬

‫ث‪ٜ‬اا ٗزغذ ؿب‪٣‬خ اُ‪ ٖ٤ٔ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪.‬‬ ‫ثؼم ذُي ص٘وبغغ ؿب‪٣‬بد اُ‪ٝ ٖ٤ٔ٤‬اُ‪٤‬ضب اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ُ٘ؾصَ ػِ‪ ٠‬شج‪ ٚ‬اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪ٗ .‬الؽع إ اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ُ‪ٜ‬ا‪ ٙ‬اُماُخ ؿ‪ٞٓ ٤‬ع‪ ٞ‬ح ‪ٝ‬ذُي الٗ٘ب ُ‪ٞ‬اخاٗب ٓضال‬ ‫‪47‬‬

‫‪Neutrosophic Science International Association‬‬ ‫‪ ε‬كِٖ ٗغم‬

‫‪ε‬‬

‫)‪(122‬‬ ‫‪ٗ ُٖٝ‬ؾصَ ػِ‪٠‬‬ ‫)‪(123‬‬ ‫‪ٝ‬ذُي الٗ‪ ٚ‬ك‪ ٢‬اُغ‪ٞ‬ا اُو ‪٣‬ت عما ٖٓ‬ ‫|‬ ‫| اًج ٖٓ اُ‪ٞ‬اؽم‪.‬‬

‫‪،‬لذ ُ‪ً ٞ‬بٗذ‬

‫|‬

‫| ص٘ؾصَ ػِ‪٠‬‬

‫رٌ‪ ٕٞ‬اُو‪ ْ٤‬أُطِوخ ُِل ‪ٝ‬م‬

‫| ‪ٝ‬‬

‫|‬

‫‪ - 2.3‬حضاب انغاَح انُ​ُىذروصىفكُح ذحهُهُا‬ ‫ُ٘ل ض إ اُـب‪٣‬خ اُزبُ‪٤‬خ ٓؼطبح‬ ‫)‪(124‬‬ ‫ُ‪ ٞ‬ػ‪ٞ‬ظ٘ب ػٖ ه‪ٔ٤‬خ ‪ x‬ثو‪ٔ٤‬خ ‪ -3‬ص٘ؾصَ ػِ‪٠‬‬

‫)‪(125‬‬ ‫‪٣‬جم‪ٝ‬ا عِ‪٤‬ب ثبٕ ‪ٛ‬ا‪ ٙ‬اُـب‪٣‬خ ُٖ رٌ‪ٓ ٕٞ‬ؼ كخ ثٔوما ؿ‪ٓ ٤‬ؼ‪ٖ٤‬‬

‫‪ٝ‬ذُي الٕ‬

‫∈ ‪.‬‬

‫ػِ‪ ٚ٤‬ص٘و‪ ّٞ‬ثزؾِ‪ َ٤‬اُجضػ ٖٓ اعَ اُزجض‪٤‬ػ ‪-:‬‬

‫)‪(126‬‬ ‫‪ُِٝ‬زبًم ٖٓ اُ٘زبئظ ثط ‪٣‬وخ اخ ‪ٓ ٟ‬ؼزٔم‪ ٖ٣‬ػِ‪ ٠‬أُؼبٓالد اُزوِ‪٤‬م‪٣‬خ (اُ‪ٜ‬رخ) ثمال ٖٓ أُؼبٓالد‬ ‫ثو‪ٔ٤‬خ – كز ح ‪ ،‬الؽع ػِ‪ ٠‬صج‪ َ٤‬أُضبٍ ال اُؾص اُؾبالد االر‪٤‬خ‪:‬‬ ‫و‪ٗ -‬بخا وًج ٗ‪ٜ‬ب‪٣‬خ صـ ‪ ٟ‬ك‪ ٢‬اُلز اد ]‪ [3,6] ٝ [1,2‬ػِ‪ ٠‬اُز‪ٞ‬اُ‪ٓ ٢‬ب ‪٣‬ؼ٘‪ ٢‬اُو‪3 ٝ 1 ْ٤‬‬ ‫ػِ‪ ٠‬اُز‪ٞ‬اُ‪. ٢‬‬

‫‪-3-1‬‬ ‫)‪(127‬‬

‫∈ ‪= -4‬‬

‫ة‪ٗ -‬بخا اصـ ٗ‪ٜ‬ب‪٣‬خ ػظٔ‪ُِ ٠‬لز اد ]‪ [3,6]ٝ [1,2‬ػِ‪ ٠‬اُز‪ٞ‬اُ‪ٓ ٢‬ب ‪٣‬ؼ٘‪ ٢‬اُو‪6 ٝ 2 ْ٤‬‬ ‫ػِ‪ ٠‬اُز‪ٞ‬اُ‪. ٢‬‬

‫= ‪-3-2‬‬ ‫)‪(128‬‬

‫∈ ‪= -5‬‬ ‫‪48‬‬

‫‪Neutrosophic Science International Association‬‬

‫د‪ٗ -‬بخا اُ٘وبغ اُ‪ٞ‬صط‪٤‬خ ُِلز اد ]‪ [3,6]ٝ [1,2‬ػِ‪ ٠‬اُز‪ٞ‬اُ‪ٓ ٢‬ب ‪٣‬ؼ٘‪ ٢‬اُو‪4.5 ٝ 1.5 ْ٤‬‬ ‫ػِ‪ ٠‬اُز‪ٞ‬اُ‪. ٢‬‬

‫‪-3‬‬ ‫∈ ‪-1.5 = -4.5‬‬

‫)‪(129‬‬ ‫س‪ -‬ثص‪ ٞ‬ح ػبٓخ ‪ٗ ،‬بخا ]‪ α ∈ [1,2‬كزٌ‪ٝ 3α ∈ [3,6] ٕٞ‬ثاُي ‪-:‬‬

‫‪-3- α‬‬ ‫)‪∈ [-3,-3]-[1,2] { since α ∈ [1,2] } = [-3-2, -3-1] = [-5, -4]. (130‬‬ ‫‪ٛٝ‬ا‪ٗ ٢ٛ ٙ‬لش اُ٘ز‪٤‬غخ اُز‪ ٢‬ؽصِ٘ب ػِ‪ٜ٤‬ب ك‪ٓ ٢‬ؼب ُخ (‪.)126‬‬

‫‪ -2.3‬انحضاب انُ​ُىذروصىفكٍ نغاَاخ انذوال انكضرَح‬ ‫√‬

‫√‬

‫√‬

‫√‬

‫)‪(131‬‬ ‫ثع ة أُوما ك‪ ٓ ٢‬اكن اُجضػ‬ ‫)‬ ‫)‬ ‫)‬

‫)‪(132‬‬

‫√(‬

‫√‬ ‫√‬

‫√(‬ ‫√(‬

‫√‬

‫√(‬

‫)‬

‫‪+‬‬

‫*‬

‫√(‬

‫)‬ ‫)‬

‫√(‬

‫و‪٣‬عب ٗضزط‪٤‬غ اُزبًم ٖٓ ه‪ٔ٤‬خ ‪ٛ‬ا‪ ٙ‬اُـب‪٣‬خ ثط ‪٣‬وخ روِ‪٤‬م‪٣‬خ ثبػزجب اُؼبَٓ ]‪ٝ α ∈ [4,5‬ؽضبة اُـب‪٣‬خ‬ ‫ث‪ٞ‬صبغخ ظ ة أُوما اٌُض ‪ ١‬ك‪ ٓ ٢‬اكن اُجضػ ‪،‬ص‪٘٤‬زظ ‪:‬‬ ‫)‪(133‬‬

‫‪∈ [4,5]/2 = [2, 2.5].‬‬

‫‪49‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ -11.3‬شثه االصرًرارَح انُ​ُىذروصىفكُح‬ ‫ص٘ومّ ‪ٝ‬ال‪ ٓ ٍٝ‬ح ٓل‪ ّٜٞ‬شج‪ – ٚ‬االصزٔ ا‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪ ،‬اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪ٓ ،‬ز‪ٓ ٠‬ب ًبٕ روبغغ اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫رٌ‪ ٕٞ‬شج‪ٓ ٚ‬ضزٔ ح ػ٘م ٗوطخ ٓؼطبح ٓضَ‬ ‫ؿ‪ ٤‬خبُ‪٤‬خ‪ ،‬و‪ ١‬وٕ‬ ‫ٖٓ اُ‪٤‬ضب ‪ ،‬اُـب‪٣‬خ اُ٘‪٤‬ز ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٖٓ اُ‪ٖ٤ٔ٤‬‬ ‫)‪(134‬‬

‫}‬

‫‪.‬‬

‫{‬

‫{‬

‫}‬

‫{‬

‫}‬

‫اذا ‪ٝ‬عمد ٗوبغ‬ ‫شج‪ٓ – ٚ‬ضزٔ ح ػِ‪ ٠‬كز ح ٓؼطبح‬ ‫‪ٝ‬اُز‪ ٌٖٔ٣ ٢‬ثط‪ٜ‬ب ثٔ٘ؾ٘‪ ٢‬روِ‪٤‬م‪ٓ ١‬ضزٔ ‪٣‬وغ اخَ اُماُخ‬

‫و‪ ١‬صزٌ‪ ٕٞ‬اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫{∈ ‪}ٝ‬‬ ‫روِ‪٤‬م‪٣‬خ }‬ ‫{∈‬ ‫‪.‬‬ ‫كعال ػٖ ذُي ‪ٗ ،‬غم لٕ اُزؼ ‪٣‬ق اُزوِ‪٤‬م‪ ٌٖٔ٣ ١‬رؼٔ‪ ٚٔ٤‬ثبُط ‪٣‬وخ اُزبُ‪٤‬خ ‪ :‬لٕ اُماُخ‬ ‫لذا رؾون‬ ‫رٌ‪ ٕٞ‬شج‪ٓ – ٚ‬ضزٔ ح ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬ب ػ٘م ًَ ٗوطخ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫)‪(135‬‬ ‫ٗغم وٕ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ أُاً‪ ٞ‬ح ك‪ ٢‬اُج٘م( ‪ٝ )1.1‬أُزٔضِخ ثبُ صْ اُج‪٤‬بٗ‪ٝ )12( ٢‬أُؼب ُخ‬ ‫‪ٝ‬ذُي ألٕ‬ ‫(‪ ٢ٛ)76‬اُخ شج‪ٓ -ٚ‬ضزٔ ح ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ػ٘م‬ ‫}‬

‫{‬

‫{‬

‫}‬

‫{‬

‫}‬

‫)‪(136‬‬

‫‪ - 11.3‬انذانح انُ​ُىذروصىفكُح انًضرًرج‬ ‫اذا ًبٕ ُغٔ‪٤‬غ‬

‫رٌ‪ٓ ٕٞ‬ضزٔ ح ػ٘م اُ٘وطخ‬

‫اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪ٞ٣ ε‬عم‬ ‫ه‪ْ٤‬‬ ‫‪ε‬‬ ‫‪,‬‬ ‫)‪(137‬‬ ‫ر‪ٞ‬عم‬ ‫∈ ‪ٌَُٝ‬‬ ‫ثؾ‪٤‬ش اٗ‪ٌَُ ٚ‬‬ ‫)‪(138‬‬ ‫(‬ ‫‪) ε‬‬ ‫‪ ٢ٛ‬ثبُؼٔ‪ٓ ّٞ‬غٔ‪ٞ‬ػخ‬ ‫ُ٘ؼ‪٤‬م اصزاًب أُؼِ‪ٓٞ‬خ اُز‪ ٢‬رو‪ ٍٞ‬إ " اُ٘وطخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ "‬ ‫‪ٓ ٢ٛٝ‬غبٓ‪٤‬غ ػ٘بص ‪ٛ‬ب اُماخِ‪٤‬خ ٓغبٓ‪٤‬غ‪.‬‬ ‫‪ٝ‬‬ ‫∈ ‪ ،‬ث‪ٔ٘٤‬ب‬

‫‪َ -12.3‬ظرَح انمًُ​ً انىصطً انُ​ُىذروصىفكُح‬ ‫ُزٌٖ‬ ‫)‪(139‬‬ ‫ؽ‪٤‬ش ‪.‬‬

‫‪⊆ ,‬‬

‫‪,‬‬

‫رٔضَ كز ح ‪ُٝ‬زٌٖ‬ ‫;‬ ‫;‬ ‫;‬ ‫‪.‬‬

‫}‬ ‫}‬ ‫}‬ ‫}‬

‫{‬ ‫{‬ ‫{‬ ‫{‬ ‫‪51‬‬

‫‪Neutrosophic Science International Association‬‬ ‫‪ٝ‬اك ض وٕ‬ ‫‪,‬‬

‫}‬

‫{‬

‫ًاُي‬ ‫{‬ ‫}‬ ‫‪.‬‬ ‫‪ ٝ‬رٔضَ ػم‬ ‫رٔضَ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ – ٚ‬ضزٔ ح ػِ‪ ٠‬كز ح ٓـِوخ‬ ‫اذا ًبٗذ‬ ‫{‬ ‫}‬ ‫∈ ثؾ‪٤‬ش إ‬ ‫ػ٘مئا ‪ٞ٣‬عم ػم ٓضَ‬ ‫‪ٝ‬اهغ ث‪ ٝ ٖ٤‬ػِٔب إ‬ ‫{ رؾز‪ ١ٞ‬اُؼم ا‪ ٝ‬إ }‬ ‫ٗوصم ثاُي إ أُغ‪ٞ‬ػخ }‬ ‫{∈ ‪.‬‬ ‫إ االصما ا‪ ٝ‬اُ٘ضخخ أُؼٔ​ٔخ ُ‪ٜ‬ا‪ ٙ‬اُ٘ظ ‪٣‬خ ‪ًٔ ٢ٛ‬ب ‪٢ِ٣‬‬ ‫〈‬ ‫‪ًٝ‬بٗذ 〉‬ ‫اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ – ٚ‬ضزٔ ح ك‪ ٢‬كز ح ٓـِوخ ٓضَ‬ ‫اذا ًبٗذ‬ ‫ك‪٢‬‬ ‫ػ٘مئا ر‪ٞ‬عم‬ ‫ػِٔب اٗ‪ٚ‬‬ ‫رٔضَ كز ح ٓؾز‪ٞ‬اح ك‪ ٢‬اُلز ح‬ ‫〈‬ ‫⊆〉‬ ‫‪ٝ‬‬ ‫ؽ‪٤‬ش إ‬ ‫ؽ‪٤‬ش إ اُص‪٤‬ـخ 〉 〈 ‪٣‬وصم ث‪ٜ‬ب و‪ٞٗ ١‬ع ٖٓ اُلز اد ٓـِوخ ًبٗذ و‪ٓ ٝ‬لز‪ٞ‬ؽخ و‪ٗ ٝ‬صق ٓـِوخ‬ ‫و‪ٗ ٝ‬صق ٓلز‪ٞ‬ؽخ ‪ًٔٝ‬ب ‪٢ِ٣‬‬ ‫‪, or‬‬ ‫‪, or‬‬ ‫‪, or‬‬ ‫‪.‬‬

‫‪ -13.3‬يصال حىل َظرَح انمًُ​ً انىصطً انُ​ُىذروصىفكُح‬ ‫ُزٌٖ‬

‫ؽ‪٤‬ش‬

‫‪ٝ‬‬

‫اُ صْ اُج‪٤‬بٗ‪)11( ٢‬‬

‫‪50‬‬

‫‪Neutrosophic Science International Association‬‬ ‫‪ُٝ ،‬زٌٖ‬

‫‪ ٢ٛ‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ػِ‪ ٠‬اُلز ح‬

‫‪,‬‬

‫}‬

‫{‬

‫ًاُي‬ ‫{‬

‫}‬ ‫}‬ ‫{‪.‬‬ ‫∈ ثؾ‪٤‬ش وٕ‬ ‫∈ ‪ ،‬ػ٘مئا ٗالؽع ‪ٝ‬ع‪ ٞ‬ه‪ ْ٤‬ػم‪٣‬مح ٍ‬ ‫‪ُٝ ،‬زٌٖ‬ ‫ػِ‪ ٠‬صج‪ َ٤‬أُضبٍ ∈‬ ‫‪٘​ٌ٘ٔ٣‬ب ؤ‪٣‬خ أُضزو‪ ْ٤‬اُؼٔ‪ ١ ٞ‬وخع اُِ‪ ٕٞ‬وػال‪، ٙ‬‬ ‫∈ ‪ٝ‬هٔ٘ب ث صْ ٓضزو‪ ْ٤‬وؽٔ اُِ‪ ٕٞ‬ػٔ‪١ ٞ‬‬ ‫‪ ،‬إ اُلٌ ح رزِخا ث ٗ‪ ٚ‬اذاًبٗذ‬ ‫‪ ،‬إ ‪ٛ‬اا أُضزو‪ ْ٤‬اُؼٔ‪ ١ ٞ‬االؽٔ ص‪٤‬زوبغغ ٓغ أُضبؽخ أُظِ​ِخ ‪ٝ‬ك‪ ٢‬ؽو‪٤‬وخ االٓ‬ ‫ٓضَ‬ ‫‪.‬‬ ‫لٕ ‪ٛ‬ا‪ ٙ‬أُ٘طوخ رٔضَ صٔب ُِماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ك‪ ٢‬اُلز ح‬

‫‪ -14.3‬يصال حىل َظرَح انمًُ​ً انىصطً انُ​ُىذروصىفكُح انًىصؼح‬ ‫ُزٌٖ‬ ‫ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ػِ‪ ٠‬اُلز ح‬ ‫ُزٌٖ‬

‫ؽ‪٤‬ش‬

‫ؽ‪٤‬ش إ‬

‫‪ٝ‬‬

‫اُخ‬

‫‪.‬‬ ‫‪,‬‬

‫}‬

‫{‬

‫‪ٝ‬‬ ‫}‬

‫‪,‬‬

‫{‬

‫‪ٗٝ‬ل ض ا‪٣‬عب‬ ‫‪.‬‬ ‫ػ٘مئا ر‪ٞ‬عم‬

‫∈‬

‫‪ٝ‬‬

‫∈‬

‫⊂‬ ‫ثؾ‪٤‬ش إ‬ ‫)‪(140‬‬

‫∈‬

‫‪.‬‬

‫اُ صْ اُج‪٤‬بٗ‪)11( ٢‬‬ ‫‪52‬‬

‫‪Neutrosophic Science International Association‬‬

‫يالحظح ‪-:‬‬ ‫إ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ذاد اُالرؼ‪ ٢ٛ ٖ٤٤‬االًض رؼو‪٤‬ما ‪ ،‬إ ٗظ ‪٣‬خ اُو‪ ٠ٔ٤‬اُ‪ٞ‬صط‪٠‬‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ اًض صؼ‪ٞ‬ثخ ‪ٝ‬رؼو‪٤‬ما ‪ٝ ،‬ك‪ ٢‬اُؾو‪٤‬وخ إ ٌَُ ص٘ق ٖٓ اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ر‪ٞ‬عم‬ ‫ٗظ ‪٣‬خ ه‪ٝ ٚٔ٤‬صط‪ٞ٤ٗ ٠‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ثص‪٤‬ـخ خبصخ ‪.‬‬ ‫‪ ًٝ‬و‪ ١‬ػبّ كبٕ ا‪ ١‬ص٘ق ٖٓ اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ر٘طجن ػِ‪ٜ٤‬ب ٗظ ‪٣‬بد ربخا ص٘لب ا‪ ٝ‬شٌال‬ ‫رخزِق ػٖ رِي اُز‪ ٢‬رزٔبش‪ٓ ٠‬غ اص٘بف اخ ‪ ٖٓ ٟ‬اُم‪ٝ‬اٍ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ‪.‬‬

‫‪ -15.3‬خىاص شثه االصرًرارَح انُ​ُىذروصىفكُح‬ ‫‪ٝ‬ذُي إ آٌٖ ثػ ٗوطخ‬ ‫رٌ‪ ٕٞ‬شج‪ٓ ٚ‬ضزٔ ح ػِ‪ ٠‬اُلز ح‬ ‫‪ -1‬إ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ‬ ‫{ ٓغ ٗوطخ ك‪ ٢‬أُغٔ‪ٞ‬ػخ }‬ ‫ك‪ ٢‬أُغٔ‪ٞ‬ػخ }‬ ‫{ ث‪ٞ‬اصطخ ٓ٘ؾ٘‪ٓ ٠‬ضزٔ روِ‪٤‬م‪ٓ ١‬ضَ‬ ‫ػِ‪ ٠‬اُلز ح‬ ‫‪ٛ ٌٕٞ٣ٝ‬اا أُ٘ؾ٘‪ٓ ٠‬ؾز‪ ٟٞ‬اخَ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ (ذاد اُضٔي) أُضٔبح‬ ‫‪.‬‬ ‫ػجب ح ػٖ ػم ؽو‪٤‬و‪ًٝ ، ٢‬بٗذ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ – ٚ‬ضزٔ ح ػ٘م‬ ‫‪ -1‬اذا ًبٗذ‬ ‫‪.‬‬ ‫‪ ٢ٛ‬و‪٣‬عب اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ - ٚ‬ضزٔ ح ػ٘م‬ ‫‪ ،‬ػ٘مئا‬ ‫اُج ‪ٛ‬بٕ‬ ‫}‬

‫}‬

‫{‬ ‫{‬

‫}‬

‫{‬

‫}‬

‫}‬

‫{(‬

‫{‬

‫{‬

‫}‬ ‫)}‬

‫)‪(141‬‬ ‫‪ٝ‬الٕ‬

‫{‬

‫ًاُي‬

‫)‪(142‬‬

‫}‬

‫‪,‬‬

‫{‬

‫ثضجت إ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ – ٚ‬ضزٔ ح‪.‬‬ ‫اُزبٕ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬زبٕ شج‪ٓ ٚ‬ضزٔ ربٕ ػ٘م‬ ‫‪ٝ‬‬ ‫‪ُ -1‬زٌٖ ًَ ٖٓ‬

‫}‬

‫{‬

‫‪ ،‬ؽ‪٤‬ش إ‬

‫ػ٘مئا ٗغم وٕ‬ ‫)‪(143‬‬

‫)(‬

‫ًِ‪ٜ‬ب ‪ٝ‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ – ٚ‬ضزٔ ح ػ٘م‬

‫‪53‬‬

‫‪Neutrosophic Science International Association‬‬ ‫اُج ا‪ٖ٤ٛ‬‬ ‫‪ ٢ٛ‬شج‪ٓ – ٚ‬ضزٔ ح ػ٘م‬

‫‪٣‬ؼ٘‪ ٢‬إ‬ ‫}‬

‫)‪(144‬‬

‫{‬

‫{‬

‫}‬

‫{‬

‫}‬

‫ُاُي كبٕ‬ ‫)‪(145‬‬

‫}‬

‫{‬

‫)‪(146‬‬

‫}‬

‫{‬

‫‪ٝ‬‬ ‫}‬

‫)‪(147‬‬ ‫ؽ‪٤‬ش إ‬

‫ًِ‪ٜ‬ب ٓغبٓ‪٤‬غ عزئ‪٤‬خ ٖٓ‬

‫‪ٝ‬ث٘لش اُؾبُخ ‪،‬‬

‫‪ٝ‬‬

‫‪ ٢ٛ‬شج‪ٓ – ٚ‬ضزٔ ح ػ٘م‬

‫)‪(148‬‬

‫ث‪ٔ٘٤‬ب‬

‫‪٣‬ؼ٘‪ ٢‬إ‬ ‫}‬

‫‪,‬‬

‫{‬

‫{‬

‫{‬

‫}‬

‫{‬

‫}‬

‫ُاُي كبٕ‬ ‫)‪(149‬‬

‫}‬

‫{‬

‫)‪(150‬‬

‫}‬

‫{‬

‫‪ٝ‬‬ ‫}‬

‫)‪(151‬‬ ‫ػِٔب إ‬

‫ًِ‪ٜ‬ب ٓغبٓ‪٤‬غ عزئ‪٤‬خ ٖٓ‬

‫‪ٝ‬‬

‫{‬

‫ث‪ٔ٘٤‬ب‬

‫االٕ‬ ‫)‪(152‬‬ ‫‪ٝ‬‬

‫‪ ٢ٛ‬شج‪ٓ – ٚ‬ضزٔ ح ػ٘م‬ ‫}‬

‫)‪(153‬‬

‫{‬

‫اذا رؾون‬ ‫{‬

‫}‬

‫{‬

‫}‬

‫ا‪ٝ‬‬ ‫}‬

‫)‪(154‬‬

‫{‬

‫{‬

‫}‬

‫{‬

‫}‬

‫ا‪ٝ‬‬ ‫)}‬ ‫)‪(155‬‬

‫{‬

‫}‬

‫{(‬

‫)}‬

‫{‬

‫}‬ ‫}‬

‫{(‬ ‫{‬

‫}‬

‫{‬ ‫‪54‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ا‪ٝ‬‬ ‫)‪(156‬‬ ‫لال وٕ ‪ٛ‬اا اُزوبغغ ُ‪٤‬ش كب ؿب ‪ٝ‬ذُي ألٗ‪ً ُٞ ٚ‬بٗذ‬ ‫‪, and‬‬ ‫∈‬ ‫)*(‬ ‫‪ٝ‬‬ ‫‪, and‬‬ ‫∈‬ ‫)**(‬ ‫‪ٝ‬اُزبُ‪ ٢‬كبٕ‬

‫∈‬ ‫∈‬ ‫∈‬

‫∈‬

‫كٕ‬

‫‪, ٝ‬‬ ‫‪, and‬‬

‫∈‬

‫‪, and‬‬

‫∈‬ ‫∈‬

‫‪,‬‬ ‫‪ٝ‬‬

‫∈‬

‫‪,‬‬ ‫‪ٝ‬‬ ‫ُاُي كبٕ‬

‫‪.‬‬ ‫‪ ٢ٛ‬ا‪٣‬عب اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ -ٚ‬ضزٔ ح ػ٘م‬

‫‪ٝ‬ثرٌَ ٓربث‪ُِ ٌٖٔ٣ ٚ‬وب ئ اصجبد إ‬

‫‪,‬‬

‫‪ٝ‬‬

‫∈‬ ‫‪.‬‬

‫ًِ‪ٜ‬ب ‪ٝ‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ ٚ‬ضزٔ ح‬

‫ٓٔب صجن ‪٘​ٌ٘ٔ٣‬ب اصز٘زبط إ‬

‫ػ٘م‬ ‫∈‬ ‫;‬ ‫)‪(157‬‬ ‫∈‬ ‫;‬ ‫)‪(158‬‬ ‫∈‬ ‫‪.‬‬ ‫)‪(159‬‬ ‫‪ ٓ .‬ح اخ ‪ٔٓٝ ، ٟ‬ب صجن‬ ‫‪ ٢ٛ‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ -ٚ‬ضزٔ ح ػ٘م‬ ‫و‪ ١‬وٕ‬ ‫ٖٓ ٓؼِ‪ٓٞ‬بد ٗغم إ‬ ‫∈‬ ‫;‬ ‫)‪(160‬‬ ‫∈‬ ‫;‬ ‫)‪(161‬‬ ‫∈‬ ‫‪.‬‬ ‫)‪(162‬‬ ‫رٔضَ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ شج‪ٓ – ٚ‬ضزٔ ح‪ٝ ،‬وخ‪ ٤‬ا ٗغم إ‬ ‫‪ٝ‬ثاُي ‪٣‬ؼ٘‪ ٢‬إ‬ ‫أُؼِ‪ٓٞ‬بد أُاً‪ ٞ‬ح ك‪ (**) ٝ (*) ٢‬رغؼِ٘ب ٗصَ ُؾو‪٤‬وخ إ ‪-:‬‬ ‫)‪(163‬‬

‫∈‬

‫)‪(164‬‬

‫∈‬

‫)‪(165‬‬

‫∈‬

‫ث‪ٜ‬اا كبٕ‬

‫) ( ‪ ٢ٛ‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ شج‪ٓ -ٚ‬ضزٔ ح ػ٘م‬

‫‪.‬‬

‫‪55‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ -16.3‬خىاص االصرًرارَح انُ​ُىذروصىفكُح‬ ‫ص٘غم ك‪ٛ ٢‬اا اُج٘م ‪ٝ‬ثط ‪٣‬وخ ٓربث‪ٜ‬خ ُِٔلب‪ ْ٤ٛ‬أُ‪ٞ‬ع‪ ٞ‬ح ك‪ ٢‬ؽضجبٕ اُزلبظَ ‪ٝ‬اُزٌبَٓ اُزوِ‪٤‬م‪، ١‬‬ ‫‪ًٝ ،‬بٗذ ∈ ػجب ح ػٖ‬ ‫‪ٝ‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ػ٘م‬ ‫اٗ‪ً ُٞ ٚ‬بٗذ‬ ‫ه‪٤‬بص‪)scalar ( ٢‬‬ ‫) ( ‪, and‬‬ ‫‪ٝ‬اُج ا‪ ٖ٤ٛ‬ثطج‪٤‬ؼخ اُؾبٍ ٗغم‪ٛ‬ب‬ ‫‪ًِٜ ،‬ب ‪ٝ‬اٍ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ػ٘م‬ ‫ؽ‪٤‬ش‬ ‫‪ٝ‬اٍ‬ ‫ٓجبش ح ًٔب ك‪ ٢‬ؽضجبٕ اُزلبظَ ‪ٝ‬اُزٌبَٓ اُزوِ‪٤‬م‪.١‬ثٔب إ ًال ٖٓ‬ ‫ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ‪ ،‬ا‪ ١‬إ‬ ‫)‪(166‬‬ ‫‪ٝ‬‬ ‫)‪(167‬‬ ‫‪ ُٞ -1‬ظ ث٘ب غ ك‪ ٢‬اُؼالهخ‬ ‫)‪(168‬‬

‫)‪ (166‬ة‬

‫ص٘ؾصَ ػِ‪٠‬‬

‫ا‪ٝ‬‬ ‫)‪(169‬‬ ‫و‪ ١‬إ‬

‫‪ ٢ٛ‬اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ػ٘م‬

‫‪.‬‬

‫‪ ُٞ -1‬اظل٘ب ؽم‪ ٝ‬اُؼالهز‪ (167)ٝ (166) ٖ٤‬ؽما ُِؾم أُوبثَ ُ‪ ٚ‬ص٘ؾصَ ػِ‪٠‬‬

‫)‪(170‬‬ ‫و‪ٝ‬‬ ‫)‪(171‬‬ ‫و‪ ١‬إ‬

‫رٔضَ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ػ٘م‬

‫‪.‬‬

‫‪ -1‬ثط ‪٣‬وخ ٓربث‪ٜ‬خ ‪ ُٞ ،‬هٔ٘ب ثط ػ )‪ (166) ٖٓ (167‬ؽما ُِؾم أُوبثَ ُ‪ ٚ‬ص٘ؾصَ ػِ‪-: ٠‬‬ ‫)‪(172‬‬ ‫و‪ٝ‬‬ ‫)‪(173‬‬ ‫و‪ ١‬إ‬

‫رٔضَ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ػ٘م‬

‫‪.‬‬ ‫‪56‬‬

‫‪Neutrosophic Science International Association‬‬ ‫‪ ُٞ -4‬ظ ث٘ب اُؼالهز‪ٓ (167) ٝ (166) ٖ٤‬غ ثؼع‪ٜٔ‬ب ( ًَ ؽم ثبُؾم أُوبثَ ُ‪ )ٚ‬ص٘ؾصَ‬ ‫ػِ‪٠‬‬ ‫]‬

‫[ ]‬

‫]‬

‫[‬

‫[‬

‫[ ]‬

‫)‪(174‬‬ ‫و‪ٝ‬‬ ‫)‪(175‬‬ ‫رٔضَ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ػ٘م‬ ‫و‪ ١‬إ‬ ‫‪ ُٞ -1‬هٔ٘ب ثوضٔخ اُؼالهز‪ ًَ (167) ، (166) ٖ٤‬ؽم ػِ‪ ٠‬اُؾم أُوبثَ ُ‪ ، ٚ‬ػِ‪ ٠‬لكز اض إ‬ ‫ُغٔ‪٤‬غ ه‪ ْ٤‬ص٘ؾصَ ػِ‪٠‬‬ ‫)‪(176‬‬

‫و‪ٝ‬‬ ‫‪+‬‬

‫)‪(177‬‬ ‫و‪ ١‬إ‬

‫) ( رٔضَ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٓضزٔ ح ػ٘م‬

‫‪ -12.3‬ذؼرَف‬

‫*‬

‫‪+‬‬

‫*‬

‫‪.‬‬

‫نهغاَاخ انُ​ُىذروصىفكُح انالَهائُح‬

‫إ اُـب‪٣‬بد اُالٗ‪ٜ‬بئ‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثبصزخماّ‬ ‫‪ًٔٝ‬ب ‪: ٢ِ٣‬‬ ‫رؼ٘‪ ٢‬وٗ‪ٌَُ ٚ‬‬ ‫و‪-‬‬ ‫}‬ ‫{ ‪.‬‬ ‫‪،‬ػ٘مئا‬ ‫رؼ٘‪ ٢‬اٗ‪ٌَُ ٚ‬‬ ‫ة‪-‬‬ ‫{‬ ‫}‬ ‫‪.‬‬ ‫‪،‬ػ٘مئا‬

‫‪٣‬ؼم ر‪ٞ‬صؼخ ُِـب‪٣‬بد اُالٗ‪ٜ‬بئ‪٤‬خ اُزوِ‪٤‬م‪٣‬خ‬ ‫‪ٞ٣‬عم‬

‫‪ ،‬ثؾ‪٤‬ش إ‬

‫‪ٞ٣‬عم‬

‫‪ ،‬ثؾ‪٤‬ش إ‬

‫‪57‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ -12.3‬أيصهح ػٍ انغاَاخ انالَهائُح انُ​ُىذروصىفكُح‬ ‫‪ُ٘ -1‬بخا اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬

‫)‪(178‬‬ ‫‪ٝ‬‬ ‫)‪(179‬‬

‫‪.‬‬

‫‪،‬‬ ‫‪ٔ٣‬ضَ أُؾبذ‪ ١‬اُؼٔ‪ُِ ١ ٞ‬ماُخ‬ ‫ُاُي‬ ‫ُطجن اُزؼ ‪٣‬ق ػِ‪ ٠‬اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ٖٓ اُ‪٤‬ضب ‪ُ ،‬زٌٖ ‪.‬‬

‫‪ ٖٓٝ‬اعَ ‪,‬‬

‫)‪(180‬‬

‫‪,‬‬

‫|‬

‫| |‬

‫|‬

‫‪ٝ‬اُز‪ ٢‬رٌبكئ‬ ‫)‪(181‬‬

‫| |‬

‫| |‬

‫ػ٘مئا‬ ‫| |‬

‫)‪(182‬‬

‫| |‬

‫ُاُي ٗغم‬ ‫)‪(183‬‬ ‫‪ُ -1‬زٌٖ‬ ‫)‪(184‬‬ ‫‪ٝ‬‬ ‫)‪(185‬‬ ‫ثاُي‬ ‫)‪(186‬‬ ‫رٔضَ ٓؾبذ‪٣‬ب ػٔ‪٣ ٞ‬ب ُِماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫ُاُي كبٕ‬ ‫‪ .‬لذا ًبٗذ‬ ‫ُؾضبة اُـب‪٣‬خ ٗلض‪ٜ‬ب‪ُ٘ .‬ل ض وٕ‬ ‫)‪(187‬‬

‫)‬

‫(‬

‫‪ُ٘ ،‬ضزخمّ االٕ رؼ ‪٣‬ق‬

‫| |‬

‫ػ٘مئا‬ ‫=‬ ‫]‬

‫)‬

‫[‬

‫(‬

‫‪58‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ؽ‪٤‬ش وٕ‬ ‫)‪(1,3)/(1,3) = (1/3, 3/1) = (1/3, 3‬‬ ‫ٓٔب ‪٣‬ؤ ‪ ١‬اُ‪ ٠‬إ أُوما اػال‪٣ ٙ‬ضب‪١ٝ‬‬ ‫)‪(188‬‬

‫= })‬

‫(‪= ( M, 12M) = M‬‬

‫(‪), and inf{M‬‬

‫‪ٝ‬ثبُزبُ‪ ٢‬كبٕ‬ ‫)‪(189‬‬ ‫‪ُ٘ -1‬ل ض وٕ‬ ‫)‪(190‬‬ ‫رٔضَ اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ( ‪ٝ‬رؼ٘‪ ٢‬اٗ٘ب ؿ‪ٓ ٤‬زبًم‪ ٖ٣‬ك‪ٔ٤‬ب اذاًبٗذ‬ ‫رٌبكئ لؽم‪ ٟ‬اُماُز‪ ٖ٤‬اُزوِ‪٤‬م‪٣‬ز‪ ٖ٤‬اُزبُ‪٤‬ز‪:ٖ٤‬‬ ‫)‪(191‬‬

‫‪.‬‬

‫) ‪ٝ‬اُز‪٢‬‬

‫ا‪ٝ‬‬

‫‪or‬‬

‫‪ٝ‬ثبُزبُ‪ ٢‬كنٕ‬ ‫)‪(192‬‬ ‫‪ٝ‬‬ ‫)‪(193‬‬ ‫‪ٔ٣‬ضالٕ ٓؾبذ‪ ٖ٤٣‬ػٔ‪ِ ٖ٤٣ ٞ‬‬ ‫و‪ٝ‬‬ ‫ُاُي‪ ،‬لٓب‬ ‫‪٘ٛ -4‬بى ٗ‪ٞ‬ع اخ ٖٓ اُـب‪٣‬بد اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ‪:ٞٛٝ‬‬

‫)‪(194‬‬ ‫‪ٝ‬‬

‫ؽ‪٤‬ش ‪٘ٛ I‬ب رٔضَ ٓ ًجخ اُالرؼ‪ٓ ٖ٤٤‬غ االخا ث٘ظ االػزجب إ‬

‫‪.‬‬

‫‪ -12.3‬دانح َُىذروصىفكُح يجانها ويجانها انًماتم يجايُغ‬ ‫‪,‬‬

‫‪,‬‬ ‫)‪(195‬‬ ‫∈ ا‪، ⊆ ٝ‬‬ ‫‪ٓ ٝ‬غبٓ‪٤‬غ ‪ٝ ،‬‬ ‫ؽ‪٤‬ش إ‬ ‫∈ ا‪ . ⊆ ٝ‬إ ص‪٤‬ـخ اُماُخ اػال‪٣ ٙ‬ؼم رؼٔ‪ٔ٤‬ب ُِماُخ اُز‪ٓ ٢‬غبُ‪ٜ‬ب ‪ٓٝ‬غبُ‪ٜ‬ب‬ ‫ًاُي‬ ‫أُوبثَ ثرٌَ كز ح ‪.‬‬ ‫ٓضبٍ‬ ‫)‪(196‬‬ ‫{‬ ‫}‬ ‫} {‬ ‫)‪(197‬‬ ‫)‪(198‬‬ ‫‪59‬‬

‫‪Neutrosophic Science International Association‬‬ ‫)‬

‫(‬

‫)‪(199‬‬ ‫} {‬ ‫‪{ }.‬‬ ‫)‪(200‬‬ ‫ٓغٔ‪ٞ‬ػخ ًَ أُغبٓ‪٤‬غ‬ ‫ٓغٔ‪ٞ‬ػخ ًَ أُغبٓ‪٤‬غ اُغزئ‪٤‬خ ٖٓ ‪ً ، M‬اُي كبٕ‬ ‫رؼم‬ ‫اُغزئ‪٤‬خ ٖٓ ‪.N‬‬ ‫‪ًٔ ،‬ب إ‬ ‫‪ٝ‬‬ ‫ًال‪ٔٛ‬ب ٓؼ ك‪ ٖ٤‬ع‪٤‬ما ػِ‪٠‬‬ ‫إ أُز ‪ ١‬اُغزئ‪ٝ ٢‬أُؼ‪٤‬ب‬ ‫رؼب ‪٣‬ق ًَ ٖٓ اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ‪ ،‬االصزٔ ا ‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ‪ ،‬أُرزوخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ‬ ‫‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ًِٜ ٢‬ب رؼٔ‪ٔ٤‬بد ا‪ ٢ٛ ٝ‬ثبالؽ ‪ ٟ‬ر‪ٞ‬ص‪٤‬ؼبد ٗلض‪ٜ‬ب ثبصزخماّ أُز ‪– ١‬‬ ‫اُغزئ‪ / ٝ ٢‬ا‪ ٝ‬أُؼ‪٤‬ب ‪.‬‬

‫‪ -21.3‬االشرماق انُ​ُىذروصىفكٍ‬ ‫إ اُزؼ ‪٣‬ق االصبص‪ ٢‬اُؼبّ ُٔرزوخ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫)‪(201‬‬

‫‪٢ٛ‬‬

‫‪.‬‬

‫ؽ‪٤‬ش إ >‪ <a, b‬رؼ٘‪ ٢‬كز ح ٓـِوخ و‪ٓ ٝ‬لز‪ٞ‬ؽخ و‪ٗ ٝ‬صق ٓلز‪ٞ‬ؽخ ‪ًٝ.‬ؾبُخ خبصخ ٖٓ اُزؼ ‪٣‬ق‬ ‫اػال‪ ٙ‬ػ٘مٓب رٌ‪ ٕٞ‬كز ح‬

‫)‪(202‬‬ ‫‪ٓ ٢ٛ‬رزوخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ِ‬ ‫ثط ‪٣‬وخ ٓجضطخ اًض ٗغم‬ ‫)‪(203‬‬ ‫ًال اُزؼ ‪٣‬ل‪ٔٛ (203) ٝ (201) ٖ٤‬ب رؼٔ‪ٔ٤‬بٕ ُزؼ ‪٣‬ق أُرزوخ اُزوِ‪٤‬م‪ ،١‬ألٗ‪ ٚ‬ك‪ ٢‬ؽو‪٤‬وخ االٓ‬ ‫ػ٘مٓب رٌ‪ ٕٞ‬اُم‪ٝ‬اٍ ‪ٝ‬أُزـ‪ ٤‬اد ًِ‪ٜ‬ب ‪ٛ‬رخ (روِ‪٤‬م‪٣‬خ) ص‪ ٌٕٞ٤‬ػ٘م‪ٛ‬ب‬ ‫)‪(204‬‬ ‫‪ٝ‬‬ ‫)‪(205‬‬ ‫‪.‬‬ ‫)‪(206‬‬ ‫ُ٘ ‪ ٟ‬ثؼط االٓضِخ األر‪٤‬خ ‪:‬‬ ‫)‪1‬‬ ‫‪.‬‬ ‫)‪(207‬‬ ‫[‬

‫]‬

‫‪+‬‬ ‫)‪(208‬‬

‫*‬ ‫‪+‬‬

‫*‬

‫‪61‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ٓؼ كخ ثبُرٌَ اُزبُ‪:٢‬‬

‫‪ُ )1‬زٌٖ‬

‫{‬

‫)‪(209‬‬

‫اُ صْ اُج‪٤‬بٗ‪)11( ٢‬‬ ‫إ و‪ ١‬اُخ روِ‪٤‬م‪٣‬خ ٓضَ‬ ‫ٓضزٔ ح ػ٘م‬

‫‪،‬‬

‫‪ ،‬ث‪ٔ٘٤‬ب‬ ‫‪ٝ‬‬

‫رٌ‪ ٕٞ‬هبثِخ ُالشزوبم ػ٘م ٗوطخ ٓضَ‬ ‫‪ًٔ ،‬ب ‪ٝ‬إ‬

‫ٓصو‪ُٞ‬خ ػ٘م‬

‫لذا رؾون ٓب ‪: ٢ِ٣‬‬ ‫ُ‪٤‬ش ُ‪ٜ‬ب ٓٔبس ػٔ‪ ١ ٞ‬ػ٘م‬ ‫} {‬

‫رٌ‪ ٕٞ‬هبثِخ ُالشزوبم ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬ب ػِ‪٠‬‬ ‫ًِ‪ٜ‬ب هبثِخ ُالشزوبم‪:‬‬

‫{‬

‫)‪(210‬‬ ‫ػ٘م‬ ‫)‪(211‬‬ ‫‪ٝ‬اال كبٕ‬

‫‪ٗ ،‬غم إ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬

‫هبثِخ ُالشزوبم اذا ًبٗذ ‪:‬‬ ‫‪,‬‬ ‫ًٔب ٗغم ك‪ ٢‬اُرٌَ اُضبثن‪ ،‬اذا ًبٗذ‬

‫ُم‪ٜ٣‬ب شج‪ٓ-ٚ‬رزوخ ػ٘م‬

‫)‪(212‬‬ ‫ا‪ ٝ‬إ‬

‫‪,‬‬ ‫ال رٌ‪ ٕٞ‬هبثِخ ُإلشزوبم ػ٘م‬

‫]‬ ‫‪ ،‬اذا ًبٗذ‬

‫‪.‬‬ ‫)‪(213‬‬ ‫]‬ ‫‪ -1‬ك‪ٔ٤‬ب ‪ٓ ٢ِ٣‬ضبٍ اخ ؽ‪ ٍٞ‬االشزوبم اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٢‬‬ ‫ُزٌٖ } {‬

‫اذا ًبٗذ‬

‫ؽ‪٤‬ش إ‬

‫[‬

‫[‬

‫رٔضَ اُالرؼ‪ٖ٤٤‬‬ ‫‪60‬‬

‫‪Neutrosophic Science International Association‬‬ ‫)‪(214‬‬

‫)‪(215‬‬ ‫ث‪ٔ٘٤‬ب ٗضزط‪٤‬غ اُؾص‪ ٍٞ‬ػِ‪ٛ ٠‬ا‪ ٙ‬اُ٘ز‪٤‬غخ ‪ٝ‬ثرٌَ ٓجبش‬ ‫)‪(216‬‬

‫‪.‬‬

‫‪ٓ -4‬ضبٍ وخ ك‪ ٚ٤‬اُالرؼ‪ٓ ٖ٤٤‬راة ‪:‬‬ ‫ٗ‪ٞ‬ع ٖٓ اٗ‪ٞ‬اع اُالرؼ‪ٗ ٖ٤٤‬ضٔ‪ ٚ٤‬اُ٘‪ٞ‬ع اال‪ٍٝ‬‬ ‫ٗضٔ‪ ٚ٤‬اُ٘‪ٞ‬ع اُضبٗ‪ ٖٓ ٢‬اُالرؼ‪ٖ٤٤‬‬ ‫ُزٌٖ‬ ‫)‪(217‬‬ ‫)‪(218‬‬ ‫ػ٘مئا‪:‬‬

‫‪{ },‬‬

‫} {‬

‫‪,‬‬

‫)‪(219‬‬

‫‪62‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ -21.3‬انركايم انُ​ُىذروصىفكٍ غُر انًحذد‬ ‫ٗؾٖ ك‪ٓ ٢‬ل‪ ّٜٞ‬اُزٌبَٓ ص٘و‪ ّٞ‬ثز‪ٞ‬ص‪٤‬غ أُل‪ ّٜٞ‬اٌُالص‪ُ ٢ٌ٤‬زؼ ‪٣‬ق ٗو‪٤‬ط –أُرزوخ (ػٌش‪-‬‬ ‫أُرزوخ )‬ ‫‪ ٢ٛ‬ا‪٣‬عب اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫إ ٗو‪٤‬ط (ػٌش) ٓرزوخ اُماُخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬ ‫‪.‬‬ ‫ثؾ‪٤‬ش إ‬ ‫كٔضال‬ ‫‪ُ -1‬زٌٖ‬ ‫)‪(220‬‬ ‫ػ٘مئا ‪،‬‬

‫} {‬

‫‪.‬‬ ‫∫‬

‫∫‬

‫∫‬ ‫)‬

‫)‪(221‬‬

‫(‬

‫∫‬

‫ثؾ‪٤‬ش إ ‪ٔ٣ C‬ضَ صبثذ ؽو‪٤‬و‪ ٢‬ؿ‪ٓ ٤‬ؼ‪ٗ ( ٖ٤‬ؼ٘‪ ٢‬ثاُي صبثذ ثبُص‪٤‬ـخ ‪ a+bI,‬ؽ‪٤‬ش إ ًَ ٖٓ ‪a,‬‬ ‫‪ ٢ٛ b‬اػما ؽو‪٤‬و‪٤‬خ ‪ ،‬ث‪ٔ٘٤‬ب ‪ٔ٣ I‬ضَ اُالرؼ‪) ٖ٤٤‬‬ ‫‪ -1‬الرؼ‪ٓ ٖ٤٤‬راة‬ ‫ُزٌٖ‬ ‫‪{ } { } { },‬‬ ‫)‪(222‬‬ ‫ًِ‪ٜ‬ب اٗ‪ٞ‬اع ٖٓ الرؼ‪٘٤٤‬بد ك ػ‪٤‬خ‬ ‫ؽ‪٤‬ش إ ‪,‬‬ ‫)‪(223‬‬ ‫ػ٘مئا‬ ‫∫‬

‫∫‬

‫)‪(224‬‬ ‫لذ إ‬

‫‪،‬‬

‫رٔضَ ص‪ٞ‬اثذ ؽو‪٤‬و‪٤‬خ ‪.‬‬

‫‪63‬‬

‫‪Neutrosophic Science International Association‬‬

‫‪ -22.3‬انركايم انُ​ُىذروصىفٍ انًحذد‬ ‫ُزٌٖ‬ ‫‪1. Let‬‬

‫)‪(225‬‬

‫اُ صْ اُج‪٤‬بٗ‪)14( ٢‬‬ ‫ثؾ‪٤‬ش إ‬ ‫)‪(226‬‬

‫{‬

‫‪.‬‬

‫ػِٔب إ‬

‫∈‬

‫ُ‪ٜ‬ب عزء ‪ٔ٣‬ضَ اُخ صٔي ‪ٝ‬ذُي ُغٔ‪٤‬غ ه‪ْ٤‬‬ ‫∈‬

‫اُخ روِ‪٤‬م‪٣‬خ ػب ‪٣‬خ ‪ٝ‬ذُي ُغٔ‪٤‬غ ه‪ْ٤‬‬

‫‪ٝ ،‬عزء اخ ‪ٔ٣‬ضَ‬

‫‪ٗ .‬ؾٖ االٕ ص٘و‪ ّٞ‬ثؾضبة اُزٌبَٓ‬

‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬أُ٘ز‪٢ٜ‬‬

‫∫‬ ‫∫‬ ‫]‬

‫∫‬

‫∫[‬

‫∫‬ ‫∫‬

‫‪+‬‬

‫∫‬ ‫∫‬

‫∫*‬

‫)‪(227‬‬ ‫ؽ‪٤‬ش ٖٓ أُؼِ‪ ّٞ‬إ‬

‫‪64‬‬

‫‪Neutrosophic Science International Association‬‬ ‫∫‬

‫‪,‬‬

‫‪,‬‬

‫∫‬

‫∫‬

‫‪,‬‬

‫)‪(228‬‬

‫‪,‬‬ ‫‪.‬‬

‫‪ ٢ٛ‬اُخ صٔي ث‪ٝ 0 ٖ٤‬‬

‫‪ ،‬ص٘ٔضَ اُ٘برظ‬

‫∫‬ ‫∫‬

‫‪and‬‬

‫ُِزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬أُ٘ز‪٢ٜ‬‬

‫‪ٝ‬ثٔب إ‬ ‫ًٔب ‪: ٢ِ٣‬‬ ‫∈‬ ‫)‪(229‬‬ ‫ًٔب ك‪ ٢‬ؽضجبٕ اُزلبظَ ‪ٝ‬اُزٌبَٓ اُزوِ‪٤‬م‪١‬‬ ‫‪ٝ‬الٗ‪ُِ ٌٖٔ٣ ٚ‬وب ئ وٕ ‪٣‬بخا‬ ‫ًٔب ‪:٢ِ٣‬‬ ‫( ٗوصم ثاُي أُضبؽخ أُ‪ٞ‬ع‪ ٞ‬ح اصلَ أُ٘ؾ٘‪ ٢‬االًض اٗخلبظب ) ا‪ ٝ‬هم رٔضَ‬ ‫)‪(230‬‬ ‫(ٗوصم ثبُٔؼمٍ ‪ ٢ٛ‬رِي أُضبؽخ اصلَ أُ٘ؾ٘‪ٓ ٢‬ب ا خالٍ ٓ٘زصق أُضبؽخ أُظِ​ِخ)‪ ،‬ا‪ ٝ‬ثٔب‬ ‫وػظْ ٓضبؽخ ٌٓٔ٘خ‬ ‫‪.‬‬ ‫)‪(231‬‬ ‫ثبالػزٔب ػِ‪ ٠‬غج‪٤‬ؼخ أُضبُخ أُؾِ‪ُٞ‬خ‪ ،‬لذ ‪٣‬ضزط‪٤‬غ اُ ‪٣‬بظ‪ ٢‬أُؾز ف ثبصزخماّ ا ‪ٝ‬اد أُ٘طن‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ ٢‬اخز‪٤‬ب ه‪ٔ٤‬خ وًض ٓالئٔخ ُٔض ُز‪. ٚ‬‬ ‫∈‬ ‫‪.‬‬ ‫)‪(232‬‬

‫‪ -23.3‬ذؼرَف يثضظ نهركايم انُ​ُىذروصىفكٍ انًحذد‬ ‫اُخ ٗ‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‬

‫ُزٌٖ‬ ‫)‪(233‬‬ ‫اُز‪ ٢‬رٌ‪ ٕٞ‬لٓب ٓضزٔ ح ا‪ ٝ‬شج‪ٓ -ٚ‬ضزٔ ح ػِ‪ ٠‬اُلز ح‬

‫ػ٘مئا‬

‫)‪(234‬‬ ‫ُغٔ‪٤‬غ ه‪}, ْ٤‬‬ ‫{∈ ‪ٝ‬‬ ‫∈‬ ‫ؽ‪٤‬ش‬ ‫‪ ،‬ثبُعجػ ًٔب ك‪٢‬‬ ‫ًِ‪ٜ‬ب ػ٘بص اخِ‪٤‬خ ‪ ،‬ا‪ ٝ‬ه‪ٞ‬اصْ ك ػ‪٤‬خ ٖٓ اُلز ح‬ ‫ثٔب رٌ‪ٓ ٕٞ‬غٔ‪ٞ‬ػخ ٖٓ اػما ؽو‪٤‬و‪٤‬خ ( ُ‪٤‬ش‬ ‫اُزؼ ‪٣‬ق اُزوِ‪٤‬م‪ُِ ١‬زٌبَٓ ‪ ،‬اال إ‬ ‫ثبُع ‪ ٝ‬ح إ رٌ‪ ٕٞ‬ص‪٤‬ـخ ‪ٛ‬ا‪ ٙ‬االػما اُؾو‪٤‬و‪٤‬خ ث٘لش ص‪٤‬ـخ رِي االػما أُؼ كخ ُم‪٘٣‬ب ك‪ ٢‬ؽضجبٕ‬ ‫ثؼعب ٖٓ اُالرؼ‪.ٖ٤٤‬‬ ‫اُزلبظَ ‪ٝ‬اُزٌبَٓ اُزوِ‪٤‬م‪ ) ١‬ا‪ ٝ‬هم رِٔي‬

‫‪ -24.3‬ذؼرَف ػاو نهركايم انُ​ُىذروصىفكٍ انًحذد‬ ‫ُزٌٖ‬ ‫)‪(235‬‬ ‫ٓ ك‪ٞ‬ػخ ُو‪ٟٞ‬‬ ‫‪ٓ ٢ٛ‬غبٓ‪٤‬غ ٖٓ‬ ‫‪ٝ‬‬ ‫رٔضَ ٓغبٓ‪٤‬غ ٓؼطبح ‪ٝ‬‬ ‫ؽ‪٤‬ش إ‬ ‫‪ ،‬كعال ػٖ إ‬ ‫‪ٓٝ‬غبُ‪ٜ‬ب أُوبثَ‬ ‫‪ ٢ٛ‬اُخ ذاد ٓغبٍ‬ ‫ٓؼ‪٘٤‬خ ػِ‪ ٠‬اُز‪ٞ‬اُ‪، ٢‬‬ ‫‪ ٢ٛ‬اُخ ذاد ٓغبٍ ‪ٓٝ‬غبٍ ٓوبثَ ػجب ح ػٖ‬ ‫‪ٛ‬ا‪ ٙ‬اُماُخ رِٔي ثؼعب ٖٓ اُالرؼ‪ٝ ٖ٤٤‬ثاُي كبٕ‬ ‫‪65‬‬

‫‪Neutrosophic Science International Association‬‬ ‫ٓغبٓ‪٤‬غ ‪ ،‬إ اُماُخ‬ ‫‪ .‬ػ٘مئا‪،‬‬ ‫∈‬

‫اُ‪ٓ ٠‬غٔ‪ٞ‬ػخ ك‪٢‬‬

‫ر٘وَ ُ٘ب ٓغٔ‪ٞ‬ػخ ٖٓ‬

‫∑‬

‫)‪(236‬‬

‫‪ُ ،‬اُي ُ‪ ٞ‬ك ظ٘ب إ‬ ‫∫‬

‫ؽ‪٤‬ش إ‬

‫‪ٗٝ‬ؾٖ ٗؼِْ إ‬ ‫∈‬ ‫ثؾ‪٤‬ش إ‬ ‫‪ٝ‬‬ ‫‪}.‬‬ ‫{ ∈ ‪, for‬‬ ‫ُاُي كبٕ اُـب‪٣‬بد اُؼِ‪٤‬ب ‪ٝ‬اُضلِ‪ُِ ٠‬زٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬ػجب ح ػٖ ٓغبٓ‪٤‬غ ( ‪٤ُٝ‬ش ثبُع ‪ ٝ‬ح‬ ‫ُغٔ‪٤‬غ ه‪ْ٤‬‬ ‫إ رٌ‪ ٕٞ‬اػما ا ‪ٛ‬رخ‪ًٔ ،‬ب ك‪ ٢‬ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُزوِ‪٤‬م‪ً ،)١‬اُي كبٕ‬ ‫}‬ ‫‪ٓ ٢ٛ‬غبٓ‪٤‬غ (‪٤ُٝ‬ش اػما ‪ٛ‬رخ ًٔب ك‪ ٢‬ؽضجبٕ اُزلبظَ‬ ‫{ ∈ ‪ًٝ‬اُي‬ ‫‪ٝ‬اُزٌبَٓ اُزوِ‪٤‬م‪ٝ )١‬ثبالظبكخ اُ‪ ٠‬ذُي ‪ ،‬ثٔب ‪٘ٛ ٌٕٞ٣‬بى ثؼط اُالرؼ‪ ٖ٤٤‬االظبك‪ ٢‬أُزؼِن ثو‪ْ٤‬‬ ‫‪ٛ‬ا‪ ٙ‬اٌُ‪ٗٞ٘٤‬بد‪.‬‬

‫‪66‬‬

‫‪Neutrosophic Science International Association‬‬

‫انفصم انراتغ‬ ‫انخاذًح‬ ‫إ اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٣ ٢‬ؼم رؼٔ‪ٔ٤‬ب ُزؾِ‪ َ٤‬أُغٔ‪ٞ‬ػخ ‪ٝ ،‬اُز‪ ٢‬ثم‪ٛ ٝ‬ب رؼم رؼٔ‪ٔ٤‬ب‬ ‫ُزؾِ‪ َ٤‬اُلز ح ‪ ،‬إ اُؾضبة اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ ٢‬اُزٔ‪٤ٜ‬م‪٣ ١‬ر‪ ٤‬اُ‪ ٠‬ه‪ ْ٤‬صبثزخ ُالرؼ‪( ٖ٤٤‬اُالرؾم‪٣‬م) ‪،‬‬ ‫ث‪ٔ٘٤‬ب ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ ٌٖٔ٣ ٢‬اػزجب ‪ٛ‬ب ‪٣‬بظ‪٤‬بد ذاد الرؼ‪ٓ ٖ٤٤‬زـ‪. ٤‬‬ ‫ٓجب ئ ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ٝ ٢‬ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٌٖٔ٣ ٢‬رط‪ٔٛ ٣ٞ‬ب ثط م ػم‪٣‬مح ‪ٝ ،‬ذُي ثبالػزٔب ػِ‪ ٠‬اٗ‪ٞ‬اع اُالرؼ‪ ٖ٤٤‬اُز‪ ٢‬رٌِٔ‪ٜ‬ب‬ ‫أُض ُخ ًاُي ثبالػزٔب ػِ‪ ٠‬اُط م أُضزخمٓخ ك‪ ٢‬اُزؼبَٓ ٓؼ‪ٜ‬ب ‪.‬‬ ‫همّ أُؤُق ك‪ٛ ٢‬اا اٌُزبة ‪ٝ‬ال‪ ٓ ٍٝ‬ح اكٌب ٌَُ ٖٓ شج‪ ٚ‬اُـب‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ‪ٝ ،‬شجخ‬ ‫االصزٔ ا ‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثط م ٓخزِلخ ػٖ شج‪ – ٚ‬االصزٔ ا ‪٣‬خ اُزوِ‪٤‬م‪٣‬خ ‪ٝ ،‬شج‪ ٚ‬أُرزوخ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪ٝ ،‬شج‪ ٚ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬ثط م رخزِق ػٖ ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ‬ ‫اُغزئ‪ ٢‬االػز‪٤‬ب ‪ ، ١‬ثبالظبكخ اُ‪ ٠‬اُزؼب ‪٣‬ق اُزوِ‪٤‬م‪٣‬خ ُِـب‪٣‬خ ‪ٝ‬االصزٔ ا ‪٣‬خ ‪ٝ‬أُرزوخ ‪ٝ‬اُزٌبَٓ‬ ‫رجبػب‪.‬‬ ‫‪ٓ ٌٖٔ٣‬ضزوجال اصخ ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬اُغزئ‪ ٖٓ ٢‬خالٍ اثؾبس‬ ‫ٓزومٓخ ك‪ٛ ٢‬اا أُعٔب ‪.‬‬ ‫همٓ٘ب ك‪ٛ ٢‬اا اٌُزبة ثؼط االٓضِخ ػٖ اُالرؼ‪٘٤٤‬بد اُخبصخ ‪ٞ٣ ٌُٖ ،‬عم اٌُض‪ٖٓ ٤‬‬ ‫اُالرؼ‪٘٤٤‬بد ك‪ ٢‬ؽ‪٤‬بر٘ب اُ‪٤ٓٞ٤‬خ ‪٣ ،‬غت ػِ‪٘٤‬ب اصز‪ٜ‬ب ‪ٝ‬اػب ح ؽِ‪ٜ‬ب ثبصزخماّ غ م ٓربث‪ٜ‬خ ا‪ ٝ‬غ م‬ ‫ٓخزِلخ ػٖ اُز‪ ٢‬ذً ٗب‪ٛ‬ب ‪ُ .‬اُي ٗؾٖ ثؾبعخ اُ‪ٓ ٠‬ز‪٣‬م ٖٓ االثؾبس اُؼِٔ‪٤‬خ ك‪ ٢‬ؽوَ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كي‪.‬‬

‫‪67‬‬

Neutrosophic Science International Association

‫شثد انًصطهحاخ‬ -AAbsolute Number Absolute Value

‫ػم ٓطِن‬ ‫ٔخ أُطِوخ‬٤‫اُو‬

Accuracy

‫هخ‬

Addition

‫عٔغ‬

Algebra

‫اُغج‬

Algorithm

‫خ‬٤ٓ‫ا ز‬ٞ‫خ‬

Amount

‫ٓوما‬

Analysis

َ٤ِ‫رؾ‬

Anti-derivative

‫ػٌش أُرزوخ‬

Anti- integration

َٓ‫ اُزٌب‬-‫ػٌش‬

Application

‫ن‬٤‫رطج‬

Applied

٢‫و‬٤‫رطج‬

Approximation

‫ت‬٣ ‫رو‬

Appurtenance Arbitrary Argument of the function= Domain

‫خ‬٤‫ أُِؾو‬-‫ اُزبثغ‬-‫أُِؾن‬ ١ ‫ب‬٤‫اخز‬ ‫ٓغبٍ اُماُخ‬

Arithmetic

‫ؽضبة‬

Asymptote

١‫ٓؾبذ‬

Axis

٢‫اؽماص‬

Axiom

)‫خ‬٤ٜ٣‫ٓضِٔخ (ثم‬

-BBasic

٢‫وصبص‬

Binary

٢‫ص٘بئ‬

Binomial 68

ٖ٣‫ اُؾم = ذاد اُؾم‬٢‫ص٘بئ‬

Neutrosophic Science International Association Bisect

‫٘صق‬٣

Boundary

ٝ‫ؽم‬

Bounded

ٝ‫ٓؾم‬

-CCalculus Call Cardinality of the set S is Center Center difference Centroid

) َٓ‫اُزٌب‬ٝ َ‫اُزٌبَٓ (ؽضجبٕ اُزلبظ‬ٝ َ‫ؽضبة اُزلبظ‬ ‫اصزمػبء‬ ‫اؽم‬ٝ ‫ٗخ ٖٓ ػ٘ص‬ٌٞٓ ‫ػخ وس‬ٞٔ‫أُغ‬ ‫ٓ ًز‬ ‫خ‬٣‫م ٓ ًز‬ٝ ‫ك‬ )‫ٌز (ٗوطخ اُزٔ ًز‬٣ ُٔ‫ا‬

Characteristic

‫ز‬٤ٔٓ

Characterizes

‫ز‬٤ٔ٣ ، ‫صق‬٣

Circumference

‫ػ‬٤‫ٓؾ‬

Classical

١‫م‬٤ِ‫رو‬

Closed

‫ٓـِن‬

Closed internal Coefficients Coincide Column Common

‫ٓغبٍ ٓـِن‬ ‫ٓؼبٓالد‬ ‫زطبثن‬٣ ٞٔ‫ػ‬ ‫ٓرز ى‬

Commutative

٢ُ‫لثما‬

Comparison

‫ٓوب ٗخ‬

Complement

ْٔ‫ٓز‬

Complex

١‫ػوم‬

Computation

‫ؽضبة‬

Condition

‫ش غ‬

Constant

‫صبثذ‬

Continuity

‫خ‬٣ ‫اصزٔ ا‬

Convergent

‫ٓزوب ة‬

Connected

َ‫ٓزص‬

69

Neutrosophic Science International Association Corresponding extremities Crisp Crisp data Curve Cross Product Cycle Cylinder

)‫االغ اف أُز٘بظ ح (االغ اف أُزٔبصِخ‬ ‫ش‬ٛ )‫خ‬٤‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ٗ ٝ‫ضذ ٓعججخ ا‬٤ُ ‫خ‬٣‫م‬٤ِ‫بٗبد رو‬٤‫رخ (ث‬ٛ ‫بٗبد‬٤‫ث‬ ‫س‬ٞ‫ – ه‬٢٘‫ٓ٘ؾ‬ ٢ٛ‫اُع ة االرغب‬ ‫ ح‬ٝ ‫اٗخ‬ٞ‫اصط‬

-DData Decimal Decreasing

‫ٓبد‬ِٞ‫ٓؼ‬ ١ ‫ػر‬ ‫ٓز٘بها‬

Define

‫ؼ ف‬٣

Definition

‫ق‬٣ ‫رؼ‬

Derivative

‫ٓرزوخ‬

Determinate Diagonal Diagram

ٖ٤‫ٓؼ‬ ‫هط‬ ٢‫ط‬٤‫صْ رخط‬

Differences

‫م‬ٝ ‫ك‬

Differential

٢ِ‫رلبظ‬

Differentiation

َ‫رلبظ‬

Dimension

‫ثؼم‬

Distribution

‫غ‬٣‫ز‬ٞ‫ر‬

discrete

‫ٓزوطغ‬

Divergent

‫ٓزجبػم‬

Dot Product

٢‫اُع ة اُ٘وط‬

-EElement

‫ػ٘ص‬

Elementary

٢ُٝ‫ا‬

71

Neutrosophic Science International Association Elimination Ellipse

‫ؽاف‬ ‫هطغ ٗبها‬

Equal

١ٝ‫ضب‬٣

Equation

‫ٓؼب ُخ‬

Equations System

‫ٗظبّ ٓؼب الد‬

Empty

‫ – كب ؽ‬٢ُ‫خب‬

Equal

١ٝ‫ٓزضب‬

Error

‫خط‬

Even

ٍ ‫ٓزؼب‬

Even function Exclusion Expansion Exponential Extrapolation

‫خ‬٤‫ع‬ٝ‫اُخ ز‬ ‫اصزجؼب‬ ‫ٗر‬ ٢‫وص‬ ٍ‫اصزٌٔب‬

-F – Factor False False position

َٓ‫ػب‬ ‫خط‬ ‫ظغ اُخبغئ‬ُٞ‫ا‬

Field

َ‫ؽو‬

Finite

ٚ‫ٓ٘ز‬

Finite Differences Formula Frontiers

‫خ‬٤ٜ‫م ٓ٘ز‬ٝ ‫ك‬ ‫ـخ‬٤‫ص‬ )‫خ‬٣ ٝ‫ّ ؽم‬ٞ‫ ( رخ‬ٝ‫ؽم‬

Function

‫اُخ‬

Functional

٢ُ‫ا‬

Fundamental

٢‫وصبص‬

-G – General term

ّ‫ؽم ػب‬

Geometric

٢‫٘مص‬ٛ

70

Neutrosophic Science International Association Global

َٓ‫شب‬

Group

‫زٓ ح‬

–H– Half Homogeneous

‫ٗصق‬ ‫ٓزغبٗش‬

Horizontal

٢‫وكو‬

Hypothesis

‫خ‬٤‫ك ظ‬

-IIdeals Implicit Inclusion isotonicity Indeterminacy

‫بد‬٤ُ‫ٓضب‬ ٢٘ٔ‫ظ‬ ٝ‫ر أُزضب‬ٞ‫اُز‬ )‫م‬٣‫ٖ (الرؾم‬٤٤‫الرؼ‬

Inequality

‫ٓز اعؾخ‬

Infinite

٢‫بئ‬ٜٗ ‫ال‬

Infimum Initial Integer Integral Intended

ٟ ‫خ صـ‬٣‫ب‬ٜٗ ‫اًج‬ ٢‫اثزمائ‬ ‫ؼ‬٤‫ػم صؾ‬ َٓ‫رٌب‬ ‫ ٓ ا‬، ٞ‫ٓوص‬

Interval

‫كز ح‬

Inverse

‫س‬ٌٞ‫ٓؼ‬

–L– Law Least-squares Limit Linear

72

ٕٞٗ‫هب‬ ٟ ‫أُ ثؼبد اُصـ‬ )‫خ‬٣‫ب‬ٜٗ) ‫خ‬٣‫ؿب‬ ٢‫خط‬

Neutrosophic Science International Association Linear Algebra

٢‫عج خط‬

Lower bound

٢ِ‫ؽم صل‬

Logarithm

ْ‫ز‬٣ ‫ؿب‬ُٞ

–M– Main

٢‫ض‬٤‫ئ‬

Map

‫ن‬٤‫رطج‬

Matrix

‫كخ‬ٞ‫ٓصل‬

Mean

‫صػ‬ٞ‫ٓز‬

Measure

‫بس‬٤‫ه‬

Mereo = Semi = Quasi

ٚ‫شج‬

Multiple Multiple- middles

‫ٓعبػق‬ ‫بد ٓزؼم ح‬٣ٞ‫ٓضز‬

-NNecessary

ّ‫الز‬

Negative

‫صبُت‬

Neutrosophic Logic

) ‫م‬٣‫ (ٓؼ كخ اُلٌ أُؾب‬٢ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫أُ٘طن ا‬

Neutrosophic Statistic

) ‫م‬٣‫ (ٓؼ كخ اُلٌ أُؾب‬٢ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫اإلؽصبء ا‬

Neutrosophic Sets

) ‫م‬٣‫خ (ٓؼ كخ اُلٌ أُؾب‬٤ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫غ ا‬٤ٓ‫أُغب‬

Neutrosophic Point Neutrosophic thick function Neutrosophic mereo- limit Nonlinear Non- discrete Norm

) ‫م‬٣‫خ (ٓؼ كخ اُلٌ أُؾب‬٤ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫اُ٘وطخ ا‬ )‫م‬٣‫خ (ٓؼ كخ اُلٌ أُؾب‬٤ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫اُخ اُضٔي ا‬ ‫خ‬٤ٌ‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ُ٘‫خ ا‬٣‫ – اُـب‬ٚ‫شج‬ ٢‫ال خط‬ ‫ ٓزوطغ‬٤‫ؿ‬ ‫ب‬٤‫ٓؼ‬

Notation

‫ ٓز‬، ‫ز‬٤ٓ ‫ر‬

Number

‫ػم‬

Numerical

73

١ ‫ػم‬

Neutrosophic Science International Association -OOpen Operator

‫ػ‬ٞ‫ٓلز‬ ‫ٓؤص‬

Openness

‫اٗلزبػ‬

Order

‫ٓ رجخ‬

Origin

َ‫ٗوطخ األص‬

Orthogonal

‫ٓزؼبٓم‬

–P– Pair

‫ط‬ٝ‫ز‬

Paradox

‫ر٘بهط‬

Parallel

١‫از‬ٞ‫ٓز‬

Partial

٢‫عزئ‬

Partial- metric Partial- Distance

٢‫ اُغزئ‬١ ‫أُز‬ ٢‫اُجؼم اُغزئ‬

Plane

ٟٞ‫ٓضز‬

Point

‫ٗوطخ‬

Polynomial

ٝ‫ ح ؽم‬٤‫ًض‬

Potentiate

َٔ‫ٓؾز‬

Pre

١‫م‬٤ٜٔ‫ر‬

-QQuadratic

٢‫ؼ‬٤‫ر ث‬

-RRadius

‫ٗصق هط‬

Range

ٟ‫ٓم‬

Rate

ٍ‫ٓؼم‬

Ratio

‫ٗضجخ‬

Rationalization

‫م‬٤‫ر ش‬

Rectangle

َ٤‫ٓضزط‬

74

Neutrosophic Science International Association Regression Relative error

‫اٗؾما‬ ٢‫خط ٗضج‬

Remainder

٢‫اُجبه‬

Restrained

‫م‬٤‫ ٓو‬، ‫م‬٤٤‫رو‬

Rounding up

‫ت‬٣ ‫رو‬

-SSet Set – argument function Set – values function Simpson's- formula

‫ػخ‬ٞٔ‫ٓغ‬ ‫غ‬٤ٓ‫ب ٓغب‬ُٜ‫اُخ ٓغب‬ ‫غ‬٤ٓ‫ب أُوبثَ ٓغب‬ُٜ‫اُخ ٓغب‬ ٕٞ‫ٔجض‬٤‫ـخ ص‬٤‫ص‬

Smooth

ٍٞ‫ ٓصو‬،ْ‫ٗبػ‬

Solution

َ‫ؽ‬

Substitution Sub- indeterminacies Supremum

‫ط‬٣ٞ‫رؼ‬ )‫خ‬٣ٞٗ‫خ (اُضب‬٤‫٘بد اُل ػ‬٤٤‫اُالرؼ‬ ٠ٔ‫خ ػظ‬٣‫ب‬ٜٗ ‫اصـ‬

Staticity = static = fixed = not moving

ًٖ‫ صب‬، ‫صبثذ‬

System of equations

‫ٗظبّ ٓؼب الد‬

-TTable

ٍٝ‫عم‬

Tangent

‫ٓٔبس‬

Technique

‫خ‬٤٘‫رو‬

Test

‫اخزجب‬

Term

‫ؽم‬

Thick Function Transcendental number

‫اُخ اُضٔي‬ ّ‫ػم ٓزضب‬

-UUnique

75

‫م‬٤‫ؽ‬ٝ

Neutrosophic Science International Association Unit Upper bound Utopian

‫ؽمح‬ٝ ١ِٞ‫ؽم ػ‬ )٢ُ‫ب‬٤‫ (خ‬٢ُ‫ٓضب‬

–V– Vague

)‫ْ (ؿبٓط‬ٜ‫ٓج‬

Value

‫ٔخ‬٤‫ه‬

Valued of the function = range Variable Vertical

‫أُغبٍ أُوبثَ ُِماُخ‬ ٤‫ٓزـ‬ ١ ٞٔ‫ػ‬

Vector

ٚ‫ٓزغ‬

Version

‫اصما‬

–W– Width

‫ػ ض‬

–Z– Zero matrix zone

76

‫خ‬٣ ‫كخ صل‬ٞ‫ٓصل‬ ‫ٗطبم‬

Neutrosophic Science International Association

‫انفصم انخايش‬ ‫انًصادر‬ ‫ انًصادر انؼرتُح‬1.5 ‫ لشأة‬, " ٢‫ك‬ٞ‫ص‬ٝ ‫ر‬ٞ٤ٗ ٞ‫خ ٖٓ ٓ٘ظ‬٤‫ صالح ب ن "اُلِضلخ اُؼ ث‬,‫ف وعلتن م عال امي‬ .1221 ,‫ ااممل عيي‬,‫ال ع عف‬

‫ انًصادر االَكهُزَح‬2.5 ‫ كرة وتحىز يُشىرج‬1.2.5 Published Papers and Books [1]

[2]

[3]

[4]

[5]

[6]

[7]

77

Agboola A.A.A., On Refined Neutrosophic Algebraic Structures, in Neutrosophic Sets and Systems, Vol. 9, 2015.

Broumi S., Smarandache F., Several Similarity Measures of Neutrosophic Sets, in Neutrosophic Sets and Systems, 54-62, Vol. 1, 2013.

Broumi S., Smarandache F., Neutrosophic Refined Similarity Measure Based on Cosine Function, in Neutrosophic Sets and Systems, 42-48, Vol. 6, 2014.

Broumi S., Smarandache F., Dhar M., Rough Neutrosophic Set, in Neutrosophic Sets and Systems, Vol. 3, 60-65, 2014.

Broumi S., Smarandache F., On Neutrosophic Implications, in Neutrosophic Sets and Systems, 9-17, Vol. 2, 2014.

Broumi S., Deli I., Smarandache F., N-Valued Interval Neutrosophic Sets and Their Application in Medical Diagnosis, in Critical Review, Center for Mathematics of Uncertainty, Creighton University, Omaha, NE, USA, Vol. X, 45-69, 2015.

Broumi S., Smarandache F., Cosine Similarity Measure of Interval Valued Neutrosophic Sets, in Neutrosophic Sets and Systems, Vol. 5, 15-20, 2014; also in Critical Review, Center for Mathematics of Uncertainty, Creighton University, USA, Vol. IX, 28-32, 2015.

Neutrosophic Science International Association Broumi S., Ye J., Smarandache F., An Extended TOPSIS Method for Multiple Attribute Decision Making based on Interval Neutrosophic Uncertain Linguistic Variables, in Neutrosophic Sets and Systems, 23-32, Vol. 8, 2015.

[9] Broumi S., Smarandache F., Interval Neutrosophic Rough Set, in Neutrosophic Sets and Systems, UNM, Vol. 7, 23-31, 2015.

[10] Broumi S., Smarandache F., Soft Interval-Valued Neutrosophic Rough Sets, in Neutrosophic Sets and Systems, UNM, Vol. 7, 69-80, 2015.

[11] Dhar M., Broumi S., Smarandache F., A Note on Square Neutrosophic Fuzzy Matrices, in Neutrosophic Sets and Systems, Vol. 3, 37-41, 2014.

[12] Farahani H., Smarandache F., Wang L. L., A Comparison of Combined Overlap Block Fuzzy Cognitive Maps (COBFCM) and Combined Overlap Block Neutrosophic Cognitive Map (COBNCM) in Finding the Hidden Patterns and Indeterminacies in Psychological Causal Models: Case Study of ADHD, in Critical Review, Center for Mathematics of Uncertainty, Creighton University, Omaha, NE, USA, Vol. X, 70-84, 2015.

[13] Kandasamy W. B. Vasantha, Smarandache F., Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps, Xiquan, Phoenix, 211 p., 2003.

[14] Kandasamy W. B. Vasantha, Smarandache F., Dual Numbers, Zip Publ., Ohio, 2012.

[15] Kandasamy W. B. Vasantha, Smarandache F., Special Dual like Numbers and Lattices, Zip. Publ., Ohio, 2012.

[16] Kandasamy W. B. Vasantha, Smarandache F., Special Quasi Dual Numbers and Groupoids, Zip Publ., 2012.

[17] Kandasamy W. B. Vasantha, Smarandache F., Neutrosophic Lattices, in Neutrosophic Sets and Systems 42-47, Vol. 2, 2014.

[18] Mukherjee A., Datta M., Smarandache F., Interval Valued Neutrosophic Soft Topological Spaces, in Neutrosophic Sets and Systems, Vol. 6, 18-27, 2014.

[19] Mumtaz Ali, Smarandache F., Shabir Muhammad, Naz Munazza, Soft Neutrosophic Bigroup and Soft Neutrosophic N-Group, in Neutrosophic Sets and Systems, 55-81, Vol. 2, 2014.

[8]

78

Neutrosophic Science International Association [20] Mumtaz

Ali, Smarandache F., Vladareanu L., Shabir M., Generalization of Soft Neutrosophic Rings and Soft Neutrosophic Fields, in Neutrosophic Sets and Systems, Vol. 6, 35-41, 2014.

[21] Mumtaz Ali, Smarandache F., Shabir M., Soft Neutrosophic Groupoids and Their Generalization, in Neutrosophic Sets and Systems, Vol. 6, 61-81, 2014.

[22] Mumtaz Ali, Smarandache F., Shabir M., Naz M., Neutrosophic BiLA-Semigroup and Neutrosophic N-LASemigroup, in Neutrosophic Sets and Systems, Vol. 4, 19-24, 2014.

[23] Mumtaz Ali, Smarandache F., Shabir M., Soft Neutrosophic Bi-LASemigroup and Soft Neutrosophic N-LA-Semigroup, in Neutrosophic Sets and Systems, Vol. 5, 45-54, 2014.

[24] Mumtaz Ali, Smarandache F., Shabir M., Vladareanu L., Generalization of Neutrosophic Rings and Neutrosophic Fields, in Neutrosophic Sets and Systems, Vol. 5, 9-14, 2014.

[25] Mumtaz Ali, Dyer C., Shabir M., Smarandache F., Soft Neutrosophic Loops and Their Generalization, in Neutrosophic Sets and Systems, Vol. 4, 55-75, 2014.

[26] Mumtaz Ali, Shabir M., Naz M., Smarandache F., Neutrosophic Left Almost Semigroup, in Neutrosophic Sets and Systems, Vol. 3, 1828, 2014.

[27] Mumtaz Ali, Smarandache F., Shabir M., Naz M., Soft Neutrosophic Ring and Soft Neutrosophic Field, in Neutrosophic Sets and Systems, Vol. 3, 53-59, 2014.

[28] Mumtaz Ali, Shabir M., Smarandache F., Vladareanu L., Neutrosophic LA-semigroup Rings, in Neutrosophic Sets and Systems, UNM, Vol. 7, 81-88, 2015.

[29] Mumtaz Ali, Smarandache F., Broumi S., Shabir M., A New Approach to Multi-Spaces through the Application of Soft Sets, in Neutrosophic Sets and Systems, UNM, Vol. 7, 34-39, 2015.

[30] Olariu S., Complex Numbers in n Dimensions, Elsevier Publication, 2002.

[31] Salama A. A., Smarandache F., Filters via Neutrosophic Crisp Sets, in Neutrosophic Sets and Systems, 34-37, Vol. 1, 2013.

[32] Salama A. A., Smarandache F., Neutrosophic Crisp Theory, in Neutrosophic Sets and Systems, Vol. 5, 27-35, 2014.

79

Neutrosophic Science International Association [33] Salama

A. A., Smarandache F., Kroumov Valeri, Neutrosophic Crisp Sets & Neutrosophic Crisp Topological Spaces, in Neutrosophic Sets and Systems, 25-30, Vol. 2, 2014.

[34] Salama A. A., Smarandache F., Eisa M., Introduction to Image Processing via Neutrosophic Technique, in Neutrosophic Sets and Systems, Vol. 5, 59-64, 2014.

[35] Salama A. A., Smarandache F., Kroumov V., Neutrosophic Closed Set and Neutrosophic Continuous Functions, in Neutrosophic Sets and Systems, Vol. 4, 4-8, 2014.

[36] Salama A. A., Smarandache F., Alblowi S. A., New Neutrosophic Crisp Topological Concept, in Neutrosophic Sets and Systems, Vol. 4, 50-54, 2014.

[37] Salama A. A., Smarandache F., Alblowi S. A., The Characteristic Function of a Neutrosophic Set, in Neutrosophic Sets and Systems, Vol. 3, 14-17, 2014.

[38] Salama A. A., El-Ghareeb H.A., Smarandache F., et. al., Introduction to Develop Some Software Programes for dealing with Neutrosophic Sets, in Neutrosophic Sets and Systems, Vol. 3, 5152, 2014.

[39] Shabir Muhammad, Mumtaz Ali, Naz Munazza, Smarandache F., Soft Neutrosophic Group, in Neutrosophic Sets and Systems, 1325, Vol. 1, 2013.

[40] Smarandache F., Neutrosophy, in Neutrosophic Probability, Set, and Logic, Amer. Res. Press, Rehoboth, USA, 105 p., 1998.

[41] Smarandache F., n-Valued Refined Neutrosophic Logic and Its Applications in Physics, in Progress in Physics, 143-146, Vol. 4, 2013.

[42] Smarandache F., Neutrosophic Measure and Neutrosophic Integral, in Neutrosophic Sets and Systems, 3-7, Vol. 1, 2013.

[43] Smarandache F., Vladutescu Stefan, Communication vs.

Information, an Axiomatic Neutrosophic Solution, in Neutrosophic Sets and Systems, 38-45, Vol. 1, 2013.

[44] Smarandache F., Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability, Sitech & Educational, Craiova, Columbus, 140 p., 2013.

[45] Smarandache F., Introduction to Neutrosophic Statistics, Sitech and Education Publisher, Craiova, 123 p., 2014.

81

Neutrosophic Science International Association [46] Smarandache

F., (t,i,f)-Neutrosophic Structures and INeutrosophic Structures, in Neutrosophic Sets and Systems, 3- 10, Vol. 8, 2015.

[47] Smarandache F., Thesis-Antithesis-Neutrothesis, and Neutrosynthesis, in Neutrosophic Sets and Systems, 64-67, Vol. 8, 2015.

[48] Smarandache F., Refined Literal Indeterminacy and the Multiplication Law of Subindeterminacies, in Neutrosophic Sets and Systems, Vol. 9, 2015.

[49] Smarandache F., Neutrosophic Axiomatic System, in Critical Review, Center for Mathematics of Uncertainty, Creighton University, Omaha, NE, USA, Vol. X, 5-28, 2015.

[50] Ye Jun, Multiple-Attribute Group Decision-Making Method under a Neutrosophic Number Environment, Journal of Intelligent Systems, DOI: 10.1515/jisys-2014-0149.

‫ مقاالت اضافية حول النيوتروسوفيك‬6.6.2 Other Articles on Neutrosophics [1]

[2]

[3]

[4]

[5]

[6]

Said Broumi, Florentin Smarandache, Correlation Coefficient of Interval Neutrosophic Set, in „Applied Mechanics and Materials”, Vol. 436 (2013), pp. 511-517, 8 p.

Said Broumi, Rıdvan Sahin, Florentin Smarandache, Generalized Interval Neutrosophic Soft Set and its Decision Making Problem, in „Journal of New Research in Science”, No. 7 (2014), pp. 29-47, 19 p.

Mumtaz Ali, Florentin Smarandache, Munazza Naz, Muhammad Shabir, G-Neutrosophic Space, in „U.P.B. Sci. Bull.”, 11 p.

Said Broumi, Irfan Deli, Florentin Smarandache, Interval Valued Neutrosophic Parameterized Soft Set Theory and its Decision Making, in „Journal of New Research in Science”, No. 7 (2014), pp. 58-71, 14 p.

Said Broumi, Florentin Smarandache, Intuitionistic Neutrosophic Soft Set, in „Journal of Information and Computing Science”, Vol. 8, No. 2, 2013, pp. 130-140, 11 p.

Said Broumi, Florentin Smarandache, Pabitra Kumar Maji, Intuitionistic Neutrosphic Soft Set over Rings, in „Mathematics and Statistics”, No. 2(3), 2014, pp.

120-126, DOI: 10.13189/ms.2014.020303, 7 p.

80

Neutrosophic Science International Association Said Broumi, Florentin Smarandache, Lower and Upper Soft Interval Valued Neutrosophic Rough Approximations of An IVNSS-Relation, at SISOM & ACOUSTICS 2014, Bucharest 22-23 May, 8 p.

[8] Said Broumi, Florentin Smarandache , More on Intuitionistic Neutrosophic Soft Sets, in „Computer Science and Information Technology”, No. 1(4), 2013, pp.

257-268, DOI: 10.13189/csit.2013.010404, 12 p.

[9] A. A. Salama, Said Broumi, Florentin Smarandache, Neutrosophic Crisp Open Set and Neutrosophic Crisp Continuity via Neutrosophic Crisp Ideals, in „I.J. Information Engineering and Electronic Business”, No. 3, 2014, pp. 1-8, DOI: 10.5815/ijieeb.2014.03.01, 8 p.

[10] Florentin Smarandache, Ştefan Vlăduţescu, Neutrosophic Principle of Interconvertibility Matter-Energy-Information, in „Journal of Information Science”, 2014, pp.

1-9, DOI: 10.1177/0165551510000000, 9 p.

[11] Florentin Smarandache, Mumtaz Ali, Munazza Naz, Muhammad Shabir, Soft Neutrosophic Left Almost Semigroup, at SISOM & ACOUSTICS 2014, Bucharest 22-23 May [12] Mumtaz Ali, Muhammad Shabir, Munazza Naz, Florentin Smarandache, Soft neutrosophic semigroups and their generalization, in „Scientia Magna”, Vol. 10 (2014), No. 1, pp. 93-111, 19 p.

[13] A. A. Salama, Said Broumi, Florentin Smarandache, Some Types of Neutrosophic Crisp Sets and Neutrosophic Crisp Relations, in „I.J. Information Engineering and Electronic Business”, 2014, 9 p.

[14] Vasile Patrascu, Neutrosophic information in the framework of multivalued representation, CAIM, Romanian Society of Applied and Industrial Mathematics et al., 19-22 September 2013, Bucharest, Romania.

[15] N-norm and N-conorm in Neutrosophic Logic and Set, and the Neutrosophic Topologies (2005), in Critical Review, Creighton University, Vol. III, 73-83, 2009.

[16] F. Smarandache, V. Christianto, n-ary Fuzzy Logic and Neutrosophic Logic Operators, in <Studies in Logic Grammar and Rhetoric>, Belarus, 17 (30), 1-16, 2009.

[17] F. Smarandache, V. Christianto, F. Liu, Haibin Wang, Yanqing Zhang, Rajshekhar Sunderraman, André Rogatko, Andrew Schumann, [7]

82

Neutrosophic Science International Association Neutrosophic Logic and Set, and Paradoxes chapters, in Multispace & Multistructure. Neutrosophic Transdisciplinarity, NESP, Finland, pp.

395-548 and respectively 604-631, 2010.

[18] Florentin Smarandache, The Neutrosophic Research Method in Scientific and Humanistic Fields, in Multispace and Multistructure, Vol. 4, 732-733, 2010.

[19] Haibin Wang, Florentin Smarandache, Yanqing Zhang, Rajshekhar Sunderraman, Single Valued Neutrosophic Sets, in Multispace and Multistructure, Vol. 4, 410-413, 2010.

[20] Pabitra Kumar Maji, Neutrosophic Soft Set, Annals of Fuzzy Mathematics and Informatics, Vol. 5, No. 1, 157-168, January 2013.

[21] Pabitra Kumar Maji, A Neutrosophic Soft Set Approach to A Decision Making Problem, Annals of Fuzzy Mathematics and Informatics, Vol. 3, No. 2, 313-319, April 2012.

[22] I. M. Hanafy, A. A. Salama, K. M. Mahfouz, Correlation Coefficients of Neutrosophic Sets by Centroid Method, , International Journal of Probability and Statistics 2013, 2(1): 9-12.

[23] Maikel Leyva-Vazquez, K. Perez-Teruel, F. Smarandache, Análisis de textos de José Martí utilizando mapas cognitivos neutrosóficos, por, 2013, http://vixra.org/abs/1303.021 [24] I. M. Hanafy, A.A.Salama and K. Mahfouz, Correlation of Neutrosophic Data, International Refereed Journal of Engineering and Science (IRJES), Vol. 1, Issue 2, 39-43, 2012.

[25] A. A. Salama & H. Alagamy, Neutrosophic Filters, International Journal of Computer Science Engineering and Information Technology Research (IJCSEITR), Vol. 3, Issue 1, Mar 2013, 307-312.

[26] Florentin Smarandache, Neutrosophic Masses & Indeterminate Models. Applications to Information Fusion, Proceedings of the 15th International Conference on Information Fusion, Singapore, 9-12 July 2012.

[27] Tzung-Pei Hong, Yasuo Kudo, Mineichi Kudo, Tsau-Young Lin, BeenChian Chien, Shyue-Liang Wang, Masahiro Inuiguchi, GuiLong Liu, A Geometric Interpretation of the Neutrosophic Set – A Generalization of the Intuitionistic Fuzzy Set, 2011 IEEE International Conference on Granular Computing, edited, IEEE Computer Society, National University of Kaohsiung, Taiwan, 602-606, 8-10 November 2011.

83

Neutrosophic Science International Association [28] Florentin

Smarandache, Luige Vladareanu, Applications of Neutrosophic Logic to Robotics / An Introduction, 2011 IEEE International Conference on Granular Computing, edited by TzungPei Hong, Yasuo Kudo, Mineichi Kudo, Tsau-Young Lin, Been-Chian Chien, Shyue-Liang Wang, Masahiro Inuiguchi, GuiLong Liu, IEEE Computer Society, National University of Kaohsiung, Taiwan, 607612, 8-10 November 2011.

[29] Said Broumi, F. Smarandache, Intuitionistic Neutrosophic Soft Set, Journal of Information and Computing Science, Vol. 8, No. 2, 2013, pp.

130-140.

[30] Wen Ju and H. D. Cheng, A Novel Neutrosophic Logic SVM (N-SVM) and its Application to Image Categorization, New Mathematics and Natural Computation (World Scientific), Vol. 9, No. 1, 27-42, 2013.

[31] A. Victor Devadoss, M. Clement Joe Anand, Activism and Nations Building in Pervasive Social Computing Using Neutrosophic Cognitive Maps (NCMs), International Journal of Computing Algorithm, Volume: 02, Pages: 257-262, October 2013.

[32] Ling Zhang, Ming Zhang, H. D. Cheng, Color Image Segmentation Based on Neutrosophic Method, in Optical Engineering, 51(3), 037009, 2012.

[33] A.Victor Devadoss, M. Clement Joe Anand, A. Joseph Bellarmin, A Study of Quality in Primary Education Combined Disjoint Block Neutrosophic Cognitive Maps (CDBNCM), Indo-Bhutan International Conference On Gross National Happiness Vol. 02, Pages: 256261,October 2013.

[34] Ming Zhang, Ling Zhang, H. D. Cheng, Segmentation of Breast Ultrasound Images Based on Neutrosophic Method, Optical Engineering, 49(11), 117001-117012, 2010.

[35] Ming Zhang, Ling Zhang, H. D. Cheng, A Neutrosophic Approach to Image Segmentation Based on Watershed Approach, Signal Processing, 90(5), 1510-1517, 2010.

[36] Florentin Smarandache, Strategy on T, I, F Operators. A Kernel Infrastructure in Neutrosophic Logic, in Multispace and Multistructure, Vol. 4, 414-419, 2010.

[37] Pinaki Majumdar & S. K. Samanta, On Similarity and Entropy of Neutrosophic Sets, M.U.C Women College, Burdwan (W. B.), India, 2013.

84

Neutrosophic Science International Association [38] Mohammad

Reza Faraji and Xiaojun Qi, An Effective Neutrosophic Set-Based Preprocessing Method for Face Recognition, Utah State University, Logan, 2013.

[39] Liu Feng, Florentin Smarandache, Toward Dialectic Matter Element of Extenics Model, in Multispace and Multistructure, Vol. 4, 420-429, 2010.

[40] Liu Feng and Florentin Smarandache, Self Knowledge and Knowledge Communication, in Multispace and Multistructure, Vol. 4, 430-435, 2010.

[41] Haibin Wang, Andre Rogatko, Florentin Smarandache, Rajshekhar Sunderraman, A Neutrosophic Description Logic, Proceedings of 2006 IEEE International Conference on Granular Computing, edited

by Yan-Qing Zhang and Tsau Young Lin, Georgia State University, Atlanta, 305-308, 2006.

[42] Haibin Wang, Rajshekhar Sunderraman, Florentin Smarandache, André Rogatko, Neutrosophic Relational Data Model, in <Critical Review> (Society for Mathematics of Uncertainty, Creighton University), Vol. II, 19-35, 2008.

[43] F. Smarandache, Short Definitions of Neutrosophic Notions [in Russian], translated by A. Schumann, Philosophical Lexicon, MinskMoscow, Econompress, Belarus-Russia, 2008.

[44] Haibin Wang, Yan-Qing Zhang, Rajshekhar Sunderraman, Florentin Smarandache, Neutrosophic Logic Based Semantic Web Services Agent, in Multispace and Multistructure, Vol. 4, 505-519, 2010.

[45] F .G. Lupiáñez, “On neutrosophic paraconsistent topology”, Kybernetes 39 (2010), 598-601.

[46] J. Ye, A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets, Journal of Intelligent and Fuzzy Systems (2013) doi: 10.3233/IFS-130916.

[47] Florentin Smarandache, Neutrosophic Logic as a Theory of Everything in Logics, in Multispace and Multistructure, Vol. 4, 525527, 2010.

[48] Florentin Smarandache, Blogs on Applications of Neutrosophics and Multispace in Sciences, in Multispace and Multistructure, Vol. 4, 528548, 2010.

[49] Athar Kharal, A Neutrosophic Multicriteria Decision Making Method, National University of Science and Technology, Islamabad, Pakistan.

85

Neutrosophic Science International Association [50] Florentin

Smarandache, Neutrosophic Transdisciplinarity (MultiSpace & Multi-Structure), Arhivele Statului, Filiala Vâlcea, Rm. Vâlcea, 1969; presented at Scoala de Vara Internationala, Interdisciplinara si Academica, Romanian Academy, Bucharest, 6-10 July 2009.

[51] Jun Ye, Single valued neutrosophic cross-entropy for multicriteria decision making problems, Applied Mathematical Modelling (2013) doi: 10.1016/j.apm.2013.07.020.

[52] Jun Ye, Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment, International Journal of General Systems, Vol. 42, No. 4, 386-394, 2013.

[53] Florentin Smarandache, Neutrosophic Diagram and Classes of Neutrosophic Paradoxes, or To The Outer-Limits of Science, Florentin Smarandache, Prog. Physics, Vol. 4, 18-23, 2010.

[54] Florentin Smarandache, S-denying a Theory, in Multi-space and Multistructure, Vol. 4, 622-629, 2010.

[55] Florentin Smarandache, Five Paradoxes and a General Question on Time Traveling, Prog. Physics, Vol. 4, 24, 2010.

[56] H. D. Cheng, Yanhui Guo and Yingtao Zhang, A Novel Image Segmentation Approach Based on Neutrosophic Set and Improved Fuzzy C-means Algorithm, New Mathematics and Natural Computation, Vol. 7, No. 1 (2011) 155-171.

[57] F. Smarandache, Degree of Negation of an Axiom, to appear in the Journal of Approximate Reasoning, arXiv:0905.0719.

[58] M. R. Bivin, N. Saivaraju and K. S. Ravichandran, Remedy for Effective Cure of Diseases using Combined Neutrosophic Relational Maps, International Journal of Computer Applications, 12(12):18?23, January 2011. Published by Foundation of Computer Science.

[59] F. Smarandache, Neutrosphic Research Method, in Multispace & Multistructure. Neutrosophic Transdiscipli-narity, NESP, Finland, pp.

395-548 and respectively 732-733, 2010.

[60] Tahar Guerram, Ramdane Maamri, and Zaidi Sahnoun, A Tool for Qualitative Causal Reasoning On Complex Systems, IJCSI International Journal of Computer Science Issues, Vol. 7, Issue 6, November 2010.

[61] P. Thiruppathi, N.Saivaraju, K.S. Ravichandran, A Study on Suicide problem using Combined Overlap Block Neutrosophic Cognitive 86

Neutrosophic Science International Association Maps, International Journal of Algorithms, Computing and Mathematics, Vol. 3, Number 4, November 2010.

[62] Francisco Gallego Lupiáñez, “On various neutrosophic topologies”, “Recent advances in Fuzzy Systems”, WSEAS (Athens , 2009), 59-62.

[63] F .G. Lupiáñez, Interval neutrosophic sets and Topology, Kybernetes 38 (2009), 621-624.

[64] F .G. Lupiáñez, On various neutrosophic topologies, Kybernetes 38 (2009), 1009-1013.

[65] Francisco Gallego Lupiáñez, Interval neutrosophic sets and topology, Kybernetes: The Intl J. of Systems & Cybernetics, Volume 38, Numbers 3-4, 2009 , pp. 621-624(4).

[66] Andrew Schumann, Neutrosophic logics on Non-Archimedean Structures, Critical Review, Creighton University, USA, Vol. III, 36-58, 2009.

[67] Fu Yuhua, Fu Anjie, Zhao Ge,Positive, Negative and Neutral Law of Universal Gravitation, Zhao Ge, New Science and Technology, 2009 (12), 30-32.

[68] Monoranjan Bhowmik and Madhumangal Pal, Intuitionistic Neutrosophic Set, Journal of Information and Computing Science, England, Vol. 4, No. 2, 2009, pp. 142-152.

[69] Wen Ju and H. D. Cheng, Discrimination of Outer Membrane Proteins using Reformulated Support Vector Machine based on Neutrosophic Set, Proceedings of the 11th Joint Conference on Information Sciences (2008), Published by Atlantis Press.

[70] Smita Rajpal, M.N. Doja, Ranjit Biswas, A Method of Imprecise Query Solving, International Journal of Computer Science and Network Security, Vol. 8 No. 6, pp. 133-139, June 2008.

[71] Florentin Smarandache, Neutrosophic Degree of a Paradoxicity, in Multispace and Multistructure, Vol. 4, 605-607, 2010.

[72] F .G. Lupiáñez, On Neutrosophic Topology, Kybernetes 37 (2008), 797-800.

[73] F .G. Lupiáñez, Interval neutrosophic sets and Topology, “Applied and Computational Mathematics”, WSEAS (Athens , 2008), 110-112.

[74] Smita Rajpal, M.N. Doja and Ranjit Biswas, A Method of Neutrosophic Logic to Answer Queries in Relational Database, by Journal of Computer Science 4 (4): 309-314, 2008.

87

Neutrosophic Science International Association [75] Pawalai

Kraipeerapun, Chun Che Fung, Kok Wai Wong, Ensemble Neural Networks Using Interval Neutrosophic Sets and Bagging, by Third International Conference on Natural Computation (ICNC 2007), Haikou, Hainan, China, August 24-August 27, 2007.

[76] Pawalai Kraipeerapun, Chun Che Fung, and Kok Wai Wong, Lithofacies Classification from Well Log Data using Neural Networks, Interval Neutrosophic Sets and Quantification of Uncertainty, World Academy of Science, Engineering and Technology, 23, 2006.

[77] Jose L. Salmeron, Florentin Smarandache, Redesigning Decision Matrix Method with an indeter-minacy-based inference process, Advances in Fuzzy Sets and Systems, Vol. 1(2), 263-271, 2006.

[78] P. Kraipeerapun, C. C. Fung, W. Brown and K. W. Wong, Neural network ensembles using interval neutrosophic sets and bagging for mineral prospectivity prediction and quantification of uncertainty, 2006 IEEE Conference on Cybernetics and Intelligent Systems, 7-9 June 2006, Bangkok, Thailand.

[79] Jose L. Salmeron, Florentin Smarandache, Processing Uncertainty and Indeterminacy in Information Systems success mapping, arXiv:cs/0512047v2.

[80] Florentin Smarandache, Jean Dezert, The Combination of Paradoxical, Uncertain, and Imprecise Sources of Information based on DSmT and Neutro-Fuzzy Inference, in arXiv:cs/0412091v1. A version of this paper published in Proceedings of 10th International Conference on Fuzzy Theory and Technology (FT&T 2005), Salt Lake City, Utah, USA, July 21-26, 2005.

[81] Goutam Bernajee, Adaptive fuzzy cognitive maps vs neutrosophic cognitive maps: decision support tool for knowledge based institution, Journal of Scientific and Industrial Research, 665-673, Vol. 67, 2008, [82] W. B. Vasantha Kandasamy and Florentin Smarandache, Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps, Book Review by Milan Mares: Kybernetika, Vol. 40 (2004), No. 1, [151]-15.

[83] H. Wang, Y. Zhang, R. Sunderraman, F. Song, Set-Theoretic Operators on Degenerated Neutrosophic Set, by Georgia State UNiversity, Atlanta, 2004.

[84] Anne-Laure Jousselme, Patrick Maupin, Neutrosophy in situation analysis, Proc. of Fusion 2004 Int. Conf. on Information Fusion, pp.

88

Neutrosophic Science International Association 400-406, Stockholm, Sweden, June 28-July1, 2004 (http://www.fusion2004.org).

[85] C. Lee, Preamble to Neutrosophic Logic, Multiple-Valued Logic / An International Journal, Vol. 8, No. 3, 285-296, June 2002.

[86] Florentin Smarandache, Neutrosophy, a New Branch of Philosophy, Multiple-Valued Logic / An International Journal, Vol. 8, No. 3, 297384, June 2002.

[87] Florentin Smarandache, A Unifying Field in Logics: Neutrosophic Field, Multiple-Valued Logic / An International Journal, Vol. 8, No. 3, 385-438, June 2002.

[88] Jean Dezert, Open Questions to Neutrosophic Inferences, MultipleValued Logic / An International Journal, Vol. 8, No. 3, 439-472, June 2002.

[89] Feng Liu, Florentin Smarandache, Logic: A Misleading Concept. A Contradiction Study toward Agent's Logic, Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics, University of New Mexico, Gallup Campus, 2001.

[90] Fu Yuhua, Fu Anjie, Zhao Ge, Six Neutral Fundamental Reactions Between Four Fundamental Reactions, by http://wbabin.net/physics/yuhua2.pdf.

[91] Florentin Smarandache, On Rugina's System of Thought, International Journal of Social Economics, Vol. 28, No. 8, 623-647, 2001.

[92] Feng Liu, Florentin Smarandache, Intentionally and Unintentionally.

On Both, A and Non-A, in Neutrosophy, Presented to the First International Conference on Neutrosophy, Neutrosophic Logic, Set, and Probability, University of New Mexico, Gallup, December 1-3, 2001.

[93] Arora, M., Biswas, R., Deployment of neutrosophic technology to retrieve answer for queries posed in natural language, Computer Science and Information Technology (ICCSIT), 2010 3rd IEEE International Conference on, Vol. 3, DOI: 10.1109/ICCSIT.2010.5564125, 2010, 435 â&#x20AC;&#x201C; 439.

[94] Aggarwal, S., Biswas, R. ; Ansari, A.Q., Neutrosophic modeling and control, Computer and Communication Technology (ICCCT), 2010 89

Neutrosophic Science International Association International Conference, DOI: 10.1109/ICCCT.2010.5640435, 2010, 718 – 723.

[95] Wang, H. ; Yan-Qing Zhang ; Sunderraman, R., Truth-value based interval neutrosophic sets, Granular Computing, 2005 IEEE International Conference, Vol. 1, DOI: 10.1109/GRC.2005.1547284, 2005, 274 – 277.

[96] Smarandache, F., A geometric interpretation of the neutrosophic set — A generalization of the intuitionistic fuzzy set, Granular Computing (GrC), 2011 IEEE International Conference, DOI: 10.1109/GRC.2011.6122665, 2011, 602 – 606.

[97] Mohan, J. ; Yanhui Guo ; Krishnaveni, V.; Jeganathan, K. MRI denoising based on neutrosophic wiener filtering, Imaging Systems and Techniques (IST), 2012 IEEE, DOI: 10.1109/IST.2012.6295518, 2012, 327 – 331.

[98] Smarandache, F. ; Vladareanu L., Applications of neutrosophic logic to robotics: An introduction, Granular Computing (GrC), 2011 IEEE, DOI: 10.1109/ GRC.2011.6122666, 2011, 607 – 612.

[99] Mohan, J. ; Krishnaveni, V. ; Guo, Yanhui, A Neutrosophic approach of MRI denoising, Image Information Processing, 2011, DOI: 10.1109/ICIIP.2011.6108880, 2011, 1 – 6.

[100] Kraipeerapun, P. ; Chun Che Fung ; Brown, W. ; Kok-Wai Wong, Neural Network Ensembles using Interval Neutrosophic Sets and Bagging for Mineral Prospectivity Prediction and Quantification of Uncertainty, Cybernetics and Intelligent Systems, 2006 IEEE Conference on, DOI: 10.1109/ICCIS.2006.252249, 2006, 1 – 6.

[101] Smarandache, F., Neutrosophic masses & indeterminate models: Applications to information fusion, Information Fusion (FUSION), 2012 15th International Conference on, 2012, 1051 – 1057.

[102] Smarandache, F., Neutrosophic set - a generalization of the intuitionistic fuzzy set, Granular Computing, 2006 IEEE, DOI: 10.1109/GRC.2006.1635754, 2006, 38 – 42.

[103] Rao, S.; Red Teaming military intelligence - a new approach based on Neutrosophic Cognitive Mapping, Intelligent Systems and Knowledge Engineering (ISKE), 2010, DOI: 10.1109/ISKE.2010.5680765, 2010, 622 – 627.

91

Neutrosophic Science International Association Smarandache, F., Neutrosophic masses & indeterminate models. Applications to information fusion, Advanced Mechatronic Systems (ICAMechS), 2012, 674 – 679.

[105] Mohan, J. ; Krishnaveni, V. ; Guo, Yanhui; Validating the Neutrosophic approach of MRI denoising based on structural similarity, Image Processing (IPR 2012), IET, DOI: 10.1049/cp.2012.0419, 2012, 1 – 6.

[106] Kraipeerapun, P. ; Chun Che Fung ; Kok Wai Wong; Ensemble Neural Networks Using Interval Neutrosophic Sets and Bagging, Natural Computation, 2007. ICNC 2007. Third International Conference, Vol. 1, DOI: 10.1109/ICNC.2007.359, 2007, 386 – 390.

[107] Kraipeerapun, P.; Chun Che Fung, Comparing performance of interval neutrosophic sets and neural networks with support vector machines for binary classification problems, Digital Ecosystems and Technologies, 2008.

DEST 2008.

2nd IEEE, DOI: 10.1109/DEST.2008.4635138, 2008, 34 – 37.

[108] Kraipeerapun, P. ; Kok Wai Wong ; Chun Che Fung ; Brown, W.; Quantification of Uncertainty in Mineral Prospectivity Prediction Using Neural Network Ensembles and Interval Neutrosophic Sets, Neural Networks, 2006.

IJCNN '06., DOI: 10.1109/IJCNN.2006.247262, 2006, 3034 – 3039.

[109] Haibin Wang; Rogatko, A.; Smarandache, F.; Sunderraman, R.; A neutrosophic description logic, Granular Computing, 2006 IEEE International Conference, DOI: 10.1109/GRC.2006.1635801, 2006, 305 – 308.

[110] Khoshnevisan, M. ; Bhattacharya, S.; Neutrosophic information fusion applied to financial market, Information Fusion, 2003.

Proceedings of the Sixth International Conference, Vol. 2, DOI: 10.1109/ICIF.2003.177381, 2003, 1252 – 1257.

[111] Aggarwal, S. ; Biswas, R. ; Ansari, A.Q. From Fuzzification to Neutrosophication: A Better Interface between Logic and Human Reasoning, Emerging Trends in Engineering and Technology (ICETET), 2010 3rd International Conference, DOI: 10.1109/ICETET.2010.26, 2010, 21 – 26.

[112] Chih-Yen Chen ; Tai-Shan Liao ; Chi-Wen Hsieh; Tzu-Chiang Liu ; Hung-Chun Chien; A novel image enhancement approach for Phalanx and Epiphyseal/metaphyseal segmentation based on hand [104]

90

Neutrosophic Science International Association radiographs, Instrumentation and Measurement Technology Conference (I2MTC), 2012 IEEE International, DOI: 10.1109/I2MTC.2012.6229651, 2012, 220-–224.

[113] Kraipeerapun, P. ; Chun Che Fung ; Kok Wai Wong, Quantification of Vagueness in Multiclass Classification Based on Multiple Binary Neural Networks, Machine Learning and Cybernetics, 2007 International Conference on, Vol. 1, DOI: 10.1109/ICMLC.2007.4370129, 2007 140 – 144.

[114] Bajger, M.; Fei Ma; Bottema, M.J.; Automatic Tuning of MST Segmentation of Mammograms for Registration and Mass Detection Algorithms, Digital Image Computing: Techniques and Applications, 2009. DICTA '09. DOI: 10.1109/DICTA.2009.72, 2009. 400 – 407.

[115] Rao, S., Externalizing Tacit knowledge to discern unhealthy nuclear intentions of nation states, Intelligent System and Knowledge Engineering, 2008. ISKE 2008. 3rd International Conference on, Vol. 1, DOI: 10.1109/ISKE.2008.4730959, 2008, 378 – 383.

[116] Maupin, P.; Jousselme, A.-L., Vagueness, a multifacet concept - a case study on Ambrosia artemisiifolia predictive cartography, Geoscience and Remote Sensing Symposium, 2004. IGARSS '04.

Proceedings.

2004 IEEE International, Vol. 1, DOI: 10.1109/IGARSS.2004.1369036, 2004.

[117] Djiknavorian, P.; Grenier, D.; Valin, P.; Analysis of information fusion combining rules under the dsm theory using ESM inputs, Information Fusion, 2007 10th International Conference on, DOI: 10.1109/ICIF.2007.4408128, 2007, 1 – 8, Cited by 4.

[118] Florentin Smarandache, A Geometric Interpretation of the Neutrosophic Set, A Generalization of the Intuitionistic Fuzzy Set, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 27-35.

[119] Hojjatollah Farahani, Florentin Smarandache, Lihshing Leigh Wang, A Comparison of Combined Overlap Block Fuzzy Cognitive Maps (COBFCM) and Combined Overlap Block Neutrosophic Cognitive Map (COBNCM) in finding the hidden patterns and indeterminacies in Psychological Causal Models: Case Study of ADHD, In Critical Review, Volume X, 2015, pp. 71-83.

92

Neutrosophic Science International Association Tudor Marin, Gheorghe Savoiu, Addressing The Dimensions Of Education And Integrated Curriculum Via Generalized Fuzzy Logic, In Euromentor Journal, Volume VI, No. 1/March 2015, pp. 61-73.

[121] T. Bharathi, A Fuzzy Study on the Analysis of Cervical Cancer among women using Combined Disjoint Block Fuzzy Cognitive Maps (CDBFCMs), In International Journal of Research in Science & Technology, Volume 1, November 2014, 5 p.

[122] Asim Hussain, Muhammad Shabir, Algebraic Structures Of Neutrosophic Soft Sets, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 53-61.

[123] Ridvan Sahin, Mesut Karabacak, A multi attribute decision making method based on inclusion measure for interval neutrosophic sets, In International Journal of Engineering and Applied Sciences, Volume 2, February 2015, pp. 13-15.

[124] Maikel Leyva-Vazquez, Karina Perez-Teruel, Florentin Smarandache, Análisis de textos de José Martí utilizando mapas cognitivos neutrosóficos, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 463-467.

[125] G. Anusha, P. Venkata Ramana, Analysis of Reasons for Stress on College Students using Combined Disjoint Block Fuzzy Cognitive Maps (CDBFCM), In International Journal For Research In Emerging Science And Technology, Volume 2, February 2015, pp. 16-21.

[126] Ștefan Vlăduțescu, Mirela Teodorescu, An analitical extended book review. S. Frunza: Advertising constructs reality (2014), In International Letters of Social and Humanistic Sciences, 2015, pp. 98106.

[127] Indranu Suhendro, An Eidetic Reflex and Moment of Breakthrough in Time and Scientific Creation: 10 Years of Progress in Physics, 100 Years of General Relativity, and the Zelmanov Cosmological Group, In Progress in Physics, Vol. 11, 2015, pp. 180182.

[128] Mumtaz Ali, Florentin Smarandache, Said Broumi , and Muhammad Shabir, A New Approach to Multi-spaces Through the Application of Soft Sets, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 34-39.

[120]

93

Neutrosophic Science International Association Anjan Mukherjee, Sadhan Sarkar, A new method of measuring similarity between two neutrosophic soft sets and its application in pattern recognition problems, In Neutrosophic Sets and Systems, Vol. 8, 2015, pp. 63-68.

[130] Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, A New Methodology for Neutrosophic Multi-Attribute Decision-Making with Unknown Weight Information, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 42-52.

[131] Fu Yuhua, An Example of Guiding Scientific Research with Philosophical Principles Based on Uniqueness of Truth and Neutrosophy Deriving Newton's Second Law and the like, In Critical Review, Volume X, 2015, pp.85-92.

[132] Said Broumi, Jun Ye, Florentin Smarandache, An Extended TOPSIS Method for Multiple Attribute Decision Making based on Interval Neutrosophic Uncertain Linguistic Variables, In Neutrosophic Sets and Systems, Vol. 8, 2015, pp.22-31.

[133] Huda E. Khalid, An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic Relation Equations (FNRE) with Geometric Programming (GP), In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 3-7.

[134] Mamouni Dhar, Said Broumi, Florentin Smarandache, A Note on Square Neutrosophic Fuzzy Matrices, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 37-41.

[135] Jun Ye, Another Form of Correlation Coefficient between Single Valued Neutrosophic Sets and Its Multiple Attribute Decision-Making Method, In Neutrosophic Sets and Systems, Vol. 1, 2013, pp. 8-12.

[136] Juan-Juan Peng, Jian-qiang Wang, Hong-yu Zhang, Xiao-hong Chen, An outranking approach for multi-criteria decision-making problemswith simplified neutrosophic sets, In Applied Soft Computing, 2014, pp. 336â&#x20AC;&#x201C;346.

[137] Yanhui Guo, Abdulkadir Sengur, A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set, In Neutrosophic Sets and Systems, Vol. 1, 2013, pp. 46-49.

[138] Yanhui Guo, Abdulkadir Sengur, Jun Ye, A novel image thresholding algorithm based on Neutro-sophic similarity score, In Measurement, 2014, pp. 175â&#x20AC;&#x201C;186.

[129]

94

Neutrosophic Science International Association Zhiming Zhang, Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incomplete weight information, In Neutrosophic Sets and Systems, Vol. 4, 2014, pp. 3549.

[140] Florentin Smarandache, Luige Vladareanu, Applications of Neutrosophic Logic to Robotics, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 61-66.

[141] Elena Rodica Opran, Dan Valeriu Voinea, Mirela Teodorescu, Art and being in neutrosophic communication, In International Letters of Social and Humanistic Sciences, 2015, pp. 16-27.

[142] C. Ramkumar, R. Ravanan, A. Lourdusamy, S. Narayanamoorthy, A Study On Neutrosophic Cognitive Maps (NCM) And Its Applications, In International Journal of Mathematical Archive, 6(3), 2015, pp. 209-211.

[143] Kalyan Mondal, Surapati Pramanik, A Study on Problems of Hijras in West Bengal Based on Neutrosophic Cognitive Maps, In Neutrosophic Sets and Systems, Vol. 5, 2014, pp. 21-26.

[144] Adrian Nicolescu, Mirela Teodorescu, A Unifying Field in Logics. Book Review, In International Letters of Social and Humanistic Sciences, 2015, pp. 48-59.

[145] A. A. Salama, Basic Structure of Some Classes of Neutrosophic Crisp Nearly Open Sets & Possible Application to GIS Topology, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 18-22.

[146] Florentin Smarandache, Stefan VladuČ&#x203A;escu, Communication vs.

Information, an Axiomatic Neutrosophic Solution, In Neutrosophic Sets and Systems, Vol. 1, 2013, pp. 38-45.

[147] Debabrata Mandal, Comparative Study of Intuitionistic and Generalized Neutrosophic Soft Sets, In International Journal of Mathematical, Computational, Natural and Physical Engineering, Vol. 9, No. 2, 2015, pp.111-114.

[148] Florentin Smarandache, Connections between Extension Logic and Refined Neutrosophic Logic, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 47-54.

[149] Said Broumi, Florentin Smarandache, Correlation Coefficient of Interval Neutrosophic Set, In Neutrosophic Theory and Its [139]

95

Neutrosophic Science International Association Applications. Collected Papers,Volume 1, EuropaNova, Bruxelles, 2014, pp. 67-73.

[150] Said Broumi, Irfan Deli, Correlation Measure For Neutrosophic Refined Sets And Its Application In Medical Diagnosis, In Palestine Journal of Mathematics, Vol. 3, 2014, pp. 11â&#x20AC;&#x201C;19.

[151] Said Broumi, Florentin Smarandache, Cosine Similarity Measure of Interval Valued Neutrosophic Sets, In Neutrosophic Sets and Systems, Vol. 5, 2014, pp. 15-20.

[152] Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Cosine Similarity Measure Based Multi-attribute Decision-making with Trapezoidal Fuzzy Neutrosophic Numbers, In Neutrosophic Sets and Systems, Vol. 8, 2014, pp. 46-55.

[153] Surapati Pramanik, Kalyan Mondal, Cosine Similarity Measure of Rough Neutrosophic Sets and Its Application In Medical Diagnosis, In Global Journal of Advanced Research, Vol. 2, pp. 315-328.

[154] Surapati Pramanik, Kalyan Mondal, Cotangent Similarity Measure of Rough Neutrosophic Sets And Its Application To Medical Diagnosis, In Journal of New Theory, 2015, pp. 90-102.

[155] Feng Liu, Florentin Smarandache, Dialectics and the Dao: On Both, A and Non-A in Neutrosophy and Chinese Philosophy, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 440-444.

[156] Shan Ye, Jun Ye, Dice Similarity Measure between Single Valued Neutrosophic Multisets and Its Application in Medical Diagnosis, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 48-53.

[157] Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments, In Neutrosophic Sets and Systems, Vol. 2, 2014, pp. 102-110.

[158] Fu Yuhua, Examples of Neutrosophic Probability in Physics, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 32-34.

[159] Fu Yuhua, Expanding Newton Mechanics with Neutrosophy and Quadstage Method. New Newton Mechanics Taking Law of Conservation of Energy as Unique Source Law, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 3-13.

96

Neutrosophic Science International Association Fu Yuhua, Expanding Uncertainty Principle to CertaintyUncertainty Principles with Neutrosophy and Quad-stage Method, In Neutrosophic Sets and Systems, Vol. 8, 2015, pp. 10-13.

[161] Mumtaz Ali, Florentin Smarandache, Muhammad Shabir, Luige Vladareanu, Generalization of Neutrosophic Rings and Neutrosophic Fields, In Neutrosophic Sets and Systems, Vol. 5, 2014, pp. 9-14.

[162] Mumtaz Ali, Florentin Smarandache, Muhammad Shabir, Luige Vladareanu, Generalization of Soft Neutrosophic Rings and Soft NeutrosophicFields, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 34-40.

[163] A. A. Salama, S. A. Alblowi, Generalized Neutrosophic Set and Generalized Neutrosophic Topological Spaces, In Computer Science and Engineering, 2012, pp. 129-132 [164] Mumtaz Ali, Florentin Smarandache, Munazza Naz, Muhammad Shabir, G-Neutrosophic Space, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 116-126.

[165] Kanika Mandal, Kajla Basu, Hypercomplex Neutrosophic Similarity Measure & Its Application In Multi-Criteria Dicision Making Problem, 15 p.

[166] Jun Ye, Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses, In Artificial Intelligence in Medicine, 2015, pp. 171â&#x20AC;&#x201C;179.

[167] Haibin Wang, Florentin Smarandache, Yan-Qing Zhang, Rajshekhar Sunderraman, Interval Neutrosophic Logic, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 142-160.

[168] A. A. Salama, Florentin Smarandache, Filters via Neutrosophic Crisp Sets, In Neutrosophic Sets and Systems, Vol. 1, 2013, pp. 34-37.

[169] Said Broumi, Florentin Smarandache, Interval Neutrosophic Rough Set, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 23-31.

[170] , I. Arockiarani, I.R. Sumathi, Interval Valued Fuzzy Neutrosophic Soft Structure Spaces, In Neutrosophic Sets and Systems, Vol. 5, 2014, pp. 36-44.

[171] Anjan Mukherjee, Mithun Datta, Florentin Smarandache, Interval Valued Neutrosophic Soft Topological Spaces, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 17-26.

[160]

97

Neutrosophic Science International Association A. A. Salama, Haitham A. El-Ghareeb, Ayman M. Manie, Florentin Smarandache, Introduction to Develop Some Software Programs for Dealing with Neutrosophic Sets, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 53-54.

[173] A. A. Salama, Florentin Smarandache, Mohamed Eisa, Introduction to Image Processing via Neutrosophic Techniques, In Neutrosophic Sets and Systems, Vol. 5, 2014, pp. 59-64.

[174] A. A. Salama, Said Broumi, S. A. Alblowi, Introduction to Neutrosophic Topological Spatial Region, Possible Application to GIS Topological Rules, In I.J. Information Engineering and Electronic Business, 2014, pp. 15-21.

[175] V. Jaiganesh, P. Rutravigneshwaran, Intrusion Detection Using Neutrosophic Classifier, In The International Journal of Science & Tech., Vol. 2, 2014, pp. 128-133.

[176] Monoranjan Bhowmik, Madhumangal Pal, Intuitionistic Neutrosophic Set Relations and Some of Its Properties, In Journal of Information and Computing Science, Vol. 5, No. 3, 2010, pp. 183-192.

[177] Broumi Said, Florentin Smarandache, Intuitionistic Neutrosophic Soft Set, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 162-171.

[178] Broumi Said, Florentin Smarandache, Intuitionistic Neutrosphic Soft Set over Rings, In Neutrosophic Theory and Its Applications.

Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 172178.

[179] Shawkat Alkhazaleh, Emad Marei, Mappings on Neutrosophic Soft Classes, In Neutrosophic Sets and Systems, Vol. 2, 2014, pp. 3-8.

[180] Shan Ye, Jing Fu, Jun Ye, Medical Diagnosis Using Distance-Based Similarity Measures of Single Valued Neutrosophic Multisets, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 47-52.

[181] Lingwei Kong, Yuefeng Wu, Jun Ye, Misfire Fault Diagnosis Method of Gasoline Engines Using the Cosine Similarity Measure of Neutrosophic Numbers, In Neutrosophic Sets and Systems, Vol. 8, 2015, pp. 42-45.

[182] Broumi Said, Florentin Smarandache, More on Intuitionistic Neutrosophic Soft Sets, In Neutrosophic Theory and Its Applications.

Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 179190.

[172]

98

Neutrosophic Science International Association Kalyan Mondal, Surapati Pramanik, Multi-criteria Group Decision Making Approach for Teacher Recruitment in Higher Education under Simplified Neutrosophic Environment, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 27-33.

[184] Yun Ye, Multiple-attribute Decision-Making Method under a Single-Valued Neutrosophic Hesitant Fuzzy Environment, In J. Intell.

Syst., 2015, pp. 23–36.

[185] Juan-Juan Peng, Multi-valued Neutrosophic Sets and Power Aggregation Operators with Their Applications in Multi-criteria Group Decision-making Problems, In International Journal of Computational Intelligence Systems, Vol. 8, No. 2, 2015, pp. 345-363.

[186] Fu Yuhua, Negating Four Color Theorem with Neutrosophy and Quadstage Method, In Neutrosophic Sets and Systems, Vol. 8, 2015, pp. 59-62.

[187] Florentin Smarandache, Neutrosofia, o nouă ramură a filosofiei, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 472-477.

[188] Florentin Smarandache, Neutrosophic Axiomatic System, In Critical Review, Volume X, 2015, pp. 5-28.

[189] Mumtaz Ali, Florentin Smarandache, Muhammad Shabir, Munazza Naz, Neutrosophic Bi-LA-Semigroup and Neutrosophic NLASemigroup, In Neutrosophic Sets and Systems, Vol. 4, 2014, pp.

19-24.

[190] Broumi Said, Florentin Smarandache, Lower and Upper Soft Interval Valued Neutrosophic Rough Approximations of An IVNSSRelation, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 191-198.

[191] Jozef Novak-Marcincin, Adrian Nicolescu, Mirela Teodorescu, Neutrosophic circuits of communication. A review, In International Letters of Social and Humanistic Sciences, 2015, pp. 174-186.

[192] A.Q. Ansari, Ranjit Biswas, Swati Aggarwal, Neutrosophic classifier: An extension of fuzzy classifer, In Applied Soft Computing, 2013, pp. 563–573.

[193] A. A. Salama, Florentin Smarandache, Valeri Kromov, Neutrosophic Closed Set and Neutrosophic Continuous Functions, In Neutrosophic Sets and Systems, Vol. 4, 2014, pp. 4-8.

[183]

99

Neutrosophic Science International Association Ameirys Betancourt-Vรกzquez, Maikel Leyva-Vรกzquez, Karina Perez-Teruel, Neutrosophic cognitive maps for modeling project portfolio interdependencies, In Critical Review, Volume X, 2015, pp.

40-44.

[195] A. A. Salama, O. M. Khaled, K. M. Mahfouz, Neutrosophic Correlation and Simple Linear Regression, In Neutrosophic Sets and Systems, Vol. 5, 2014, pp. 3-8.

[196] A. A. Salama, Said Broumi, Florentin Smarandache, Neutrosophic Crisp Open Set and Neutrosophic Crisp Continuity via Neutrosophic Crisp Ideals, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp 199-205.

[197] A. A. Salama, Neutrosophic Crisp Points & Neutrosophic Crisp Ideals, In Neutrosophic Sets and Systems, Vol. 1, 2013, pp. 50-53.

[198] A. A. Salama, Hewayda Elghawalby, *- Neutrosophic Crisp Set & *Neutrosophic Crisp relations, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 12-16.

[199] A. A. Salama, Florentin Smarandache, Valeri Kroumov, Neutrosophic Crisp Sets & Neutrosophic Crisp Topological Spaces, In Neutrosophic Sets and Systems, Vol. 2, 2014, pp. 25-30.

[200] A. A. Salama, Florentin Smarandache, Neutrosophic Crisp Set Theory, In Neutrosophic Sets and Systems, Vol. 5, 2014, pp. 27-35.

[201] Kalyan Mondal, Surapati Pramanik, Neutrosophic Decision Making Model of School Choice, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 62-68.

[202] A. A. Salama, H. Alagamy, Neutrosophic Filters, In International Journal of Computer Science Engineering and Information Technology Research (IJCSEITR), Vol. 3, 2013, pp. 307-312.

[203] Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir, In Neutrosophic Sets and Systems, Vol. 2, 2013, pp. 82-100.

[204] Ridvan Sahin, Neutrosophic Hierarchical Clustering Algoritms, In Neutrosophic Sets and Systems, Vol. 2, 2014, pp. 18-24.

[205] A.A.A. Agboola, S.A. Akinleye, Neutrosophic Hypercompositional Structures defined by Binary Relations, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 29-36.

[206] , A.A.A. Agboola, S.A. Akinleye, Neutrosophic Hypervector Spaces, 16 p.

[194]

011

Neutrosophic Science International Association A. A. Salama, Florentin Smarandache, Neutrosophic Ideal Theory Neutrosophic Local Function and Generated Neutrosophic Topology, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 213-218.

[208] Mumtaz Ali, Muhammad Shabir, Florentin Smarandache, Luige Vladareanu, Neutrosophic LA-Semigroup Rings, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 81-88.

[209] Vasantha Kandasamy, Florentin Smarandache, Neutrosophic Lattices, In Neutrosophic Sets and Systems, Vol. 2, 2013, pp. 42-47.

[210] Mumtaz Ali, Muhammad Shabir, Munazza Naz, Florentin Smarandache, Neutrosophic Left Almost Semigroup, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 18-28.

[211] Alexandru Gal, Luige Vladareanu, Florentin Smarandache, Hongnian Yu, Mincong Deng, Neutrosophic Logic Approaches Applied to ”RABOT” Real Time Control, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 55-60.

[212] Karina Pérez-Teruel, Maikel Leyva-Vázquez, Neutrosophic Logic for Mental Model Elicitation and Analysis, In Neutrosophic Sets and Systems, Vol. 2, 2014, pp. 31-33.

[213] Fu Yuhua, Neutrosophic Examples in Physics, In Neutrosophic Sets and Systems, Vol. 1, 2013, pp. 26-33.

[214] Florentin Smarandache, Neutrosophic Measure and Neutrosophic Integral, In Neutrosophic Sets and Systems, Vol. 1, 2013, pp. 3-7.

[215] Swati Aggarwal, Ranjit Biswas, A.Q. Ansari, Neutrosophic Modeling and Control, Intl. Conf. on Computer & Communication Tech., 2010, pp. 718-723.

[216] Irfan Deli, Yunus Toktas, Said Broumi, Neutrosophic Parameterized Soft Relations and Their Applications, In Neutrosophic Sets and Systems, Vol. 4, 2014, pp. 25-34.

[217] Said Broumi, Irfan Deli, Florentin Smarandache, Neutrosophic Parametrized Soft Set Theory and Its Decision Making, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 403-409.

[218] Florentin Smarandache, Stefan Vladutescu, Neutrosophic Principle of Interconvertibility Matter-Energy-Information [207]

010

Neutrosophic Science International Association (NPI_MEI), In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 219-227.

[219] Said Broumi, Irfan Deli, Florentin Smarandache, Neutrosophic Refined Relations and Their Properties, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 228-248.

[220] Said Broumi, Florentin Smarandache, Neutrosophic Refined Similarity Measure Based on Cosine Function, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 41-47.

[221] Kalyan Mondal, Surapati Pramanik, Neutrosophic Refined Similarity Measure Based On Cotangent Function And Its Application To Multi-Attribute Decision Making, In Global Journal of Advanced Research, Vol-2, 2015, pp. 486 -496.

[222] A. A. Salama, Mohamed Eisa, M. M. Abdelmoghny, Neutrosophic Relations Database, In International Journal of Information Science and Intelligent System, 2014, pp. 1-13.

[223] Daniela Gifu, Mirela Teodorescu, Neutrosophic routes in multiverse of communication, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 81-83.

[224] A.A.Salama, S.A. Alblowi, Neutrosophic Set and Neutrosophic Topological Spaces, In IOSR Journal of Mathematics, 2012, pp. 31-35.

[225] Mehmet Sahin, Shawkat Alkhazaleh, Vakkas Ulucay, Neutrosophic Soft Expert Sets, In Applied Mathematics, 2015, pp. 116-127.

[226] Irfan Deli, Said Broumi, Neutrosophic soft matrices and NSMdecision making, In Journal of Intelligent & Fuzzy Systems, 2015, pp.

2233â&#x20AC;&#x201C;2241.

[227] Irfan Deli, Said Broumi, Mumtaz Ali, Neutrosophic Soft Multi-Set Theory and Its Decision Making, In Neutrosophic Sets and Systems, Vol. 5, 2014, pp. 65-76.

[228] Irfan Deli, Said Broumi, Neutrosophic soft relations and some properties, In Ann. Fuzzy Math. Inform., 2014, pp. 2-14.

[229] Debabrata Mandal, Neutrosophic Soft Semirings, In Annals of Fuzzy Mathematics and Informatics, 2014, pp. 2-13.

[230] Faruk Karaaslan, Neutrosophic Soft Sets with Applications in Decision Making, In International Journal of Information Science and Intelligent System, 2015, pp. 1-20.

012

Neutrosophic Science International Association Shawkat Alkhazaleh, Neutrosophic Vague Set Theory, In Critical Review, Volume X, 2015, pp. 29-39.

[232] A.A.A. Agboola, S.A. Akinleye, Neutrosophic Vector Spaces, In Neutrosophic Sets and Systems, Vol. 4, 2014, pp. 9-18.

[233] Said Broumi, Florentin Smarandache, New Distance and Similarity Measures of Interval Neutrosophic Sets, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 249-255.

[234] A. A. Salama, Florentin Smarandache, S. A. Alblowi, New Neutrosophic Crisp Topological Concepts, In Neutrosophic Sets and Systems, Vol. 4, 2014, pp. 50-54.

[235] I. R. Sumathi, I. Arockiarani, New operations On Fuzzy Neutrosophic Mattrices, In International Journal of Innovative Research and study, 2014, pp. 119-124.

[236] Said Broumi, Florentin Smarandache, New Operations on Interval Neutrosophic Sets, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 256-266.

[237] Said Broumi, Pinaki Majumdar, Florentin Smarandache, New Operations on Intuitionistic Fuzzy Soft Sets Based on First Zadeh's Logical Operators, In Neutrosophic Theory and Its Applications.

Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 267277.

[238] Said Broumi, Florentin Smarandache, New Operations over Interval Valued Intuitionistic Hesitant Fuzzy Set, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp 267-276.

[239] Said Broumi, Florentin Smarandache, Mamoni Dhar, Pinaki Majumdar, New Results of Intuitionistic Fuzzy Soft Set, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 386-391.

[240] Vasantha Kandasamy, Sekar. P. Vidhyalakshmi, New Type of Fuzzy Relational Equations and Neutrosophic Relational Equations â&#x20AC;&#x201C; To analyse Customers Preference to Street Shops, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 68-76.

[241] Irfan Deli, npn-Soft sets theory and their applications, In Annals of Fuzzy Mathematics and Informatics, 2015, pp. 3-16.

[231]

013

Neutrosophic Science International Association Said Broumi, Irfan Deli, Florentin Smarandache, N-Valued Interval Neutrosophic Sets and Their Application in Medical Diagnosis, In Critical Review, Volume X, 2015, pp. 45-68.

[243] Florentin Smarandache, n-Valued Refined Neutrosophic Logic and Its Applications to Physics, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 36-44.

[244] Said Broumi, Florentin Smarandache, Mamoni Dhar, On Fuzzy Soft Matrix Based on Reference Function, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 392-398.

[245] Tanushree Mitra Basu, Shyamal Kumar Mondal, Neutrosophic Soft Matrix and Its Application in Solving Group Decision Making Problems from Medical Science, In Computer Communication & Collaboration, 2015, Vol. 3, pp. 1-31.

[246] A.A.A. Agboola, B. Davvaz, On Neutrosophic Canonical Hypergroups and Neutro-sophic Hyperrings, In Neutrosophic Sets and Systems, Vol. 2, 2014, pp. 34-41.

[247] A.A.A. Agboola, B. Davvaz, On Neutrosophic Ideals of Neutrosophic BCI-Algebras, In Critical Review, Volume X, 2015, pp. 93-103.

[248] Fu Yuhua, Pauli Exclusion Principle and the Law of Included Multiple-Middle, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp.

3-5.

[249] Pawalai Krai Peerapun, Kok Wai Wong, Chun Che Fung, Warick Brown, Quantification of Uncertainty in Mineral Prospectivity Prediction Using Neural Network Ensembles and Interval Neutrosophic Sets, 2006 International Joint Conference on Neural Networks, pp. 3034-3039.

[250] Florentin Smarandache, Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies, In Neutrosophic Sets and Systems, Vol. 9, 2015, pp. 1-5.

[251] Said Broumi, Irfan Deli, Florentin Smarandache, Relations on Interval Valued Neutrosophic Soft Sets, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 290-306.

[252] Florentin Smarandache, Reliability and Importance Discounting of Neutrosophic Masses, In Neutrosophic Theory and Its [242]

014

Neutrosophic Science International Association Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 13-26.

[253] Florentin Smarandache, Replacing the Conjunctive Rule and Disjunctive Rule with Tnorms and T-conorms respectively (Tchamova-Smaran-dache), in Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 45-46.

[254] Said Broumi, Florentin Smarandache, On Neutrosophic Implications, In Neutrosophic Sets and Systems, Vol. 2, 2014, pp. 917.

[255] A. A. Salama, Mohamed Eisa, S.A. Elhafeez, M. M. Lotfy, Review of Recommender Systems Algorithms Utilized in Social Networks based e-Learning Systems & Neutrosophic System, In Neutrosophic Sets and Systems, Vol. 8, 2015, pp. 32-40.

[256] Kalyan Mondal, Surapati Pramanik, Rough Neutrosophic MultiAttribute Decision-Making Based on Rough Accuracy Score Function, In Neutrosophic Sets and Systems, Vol. 8, 2015, pp. 14-21.

[257] Said Broumi, Florentin Smarandache, Mamoni Dhar, Rough Neutrosophic Sets, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 62-67.

[258] C. Antony Crispin Sweety, I. Arockiarani, Rough sets in Fuzzy Neutrosophic approximation space, 16 p.

[259] Said Broumi, Florentin Smarandache, Several Similarity Measures of Neutrosophic Sets, In Neutrosophic Sets and Systems, Vol. 1, 2013, pp. 54-62.

[260] Anjan Mukherjee and Sadhan Sarkar, Several Similarity Measures of Interval Valued Neutrosophic Soft Sets and Their Application in Pattern Recognition Problems, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 54-60.

[261] Zhang-peng Tian, Jing Wang, Hong-yu Zhang, Xiao-hong Chen, Jian-qiang Wang, Simplified neutrosophic linguistic normalized weighted Bonferroni mean operator and its application to multicriteria decision-making problems, Faculty of Sciences and Mathematics, University of Nis, Serbia, Filomat, 24 p.

[262] Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making, In Neutrosophic Sets and Systems, Vol. 2, 2014, pp. 48-54.

015

Neutrosophic Science International Association Rajashi Chatterjee, P. Majumdar, S. K. Samanta, Single valued neutrosophic multisets, In Annals of Fuzzy Mathematics and Informatics, 2015, pp. 1-16.

[264] Said Broumi, Florentin Smarandache, Soft Interval â&#x20AC;&#x201C;Valued Neutrosophic Rough Sets, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 69-80.

[265] Mumtaz Ali, Florentin Smarandache, Muhammad Shabir, Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutrosophic N-Group, In Neutrosophic Sets and Systems, Vol. 2, 2014, pp. 55-79.

[266] Mumtaz Ali, Florentin Smarandache, Muhammad Shabir, Soft Neutrosophic Bi-LA-semigroup and Soft Neutrosophic N-LAseigroup, In Neutrosophic Sets and Systems, Vol. 5, 2014, pp. 45-58.

[267] Muhammad Shabir, Mumtaz Ali, Munazza Naz, Florentin Smarandache, Soft Neutrosophic Group, In Neutrosophic Sets and Systems, Vol. 1, 2013, pp. 13-25.

[268] Mumtaz Ali, Florentin Smarandache, Muhammad Shabir, Soft Neutrosophic Groupoids and Their Generalization, In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 61-80.

[269] Florentin Smarandache, Mumtaz Ali, Munazza Naz, and Muhammad Shabir, Soft Neutrosophic Left Almost Semigroup, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 317-326.

[270] Mumtaz Ali, Florentin Smarandache, and Muhammad Shabir, Soft Neutrosophic Loop, Soft Neutrosophic Biloop and Soft Neutrosophic N-Loop, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 327-348.

[271] Mumtaz Ali, Christopher Dyer, Muhammad Shabir, Florentin Smarandache, Soft Neutrosophic Loops and Their Generalization, In Neutrosophic Sets and Systems, Vol. 4, 2014, pp. 55-75.

[272] Mumtaz Ali, Florentin Smarandache, Muhammad Shabir, Munazza Naz, Soft Neutrosophic Ring and Soft Neutrosophic Field, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 55-61.

[273] Mumtaz Ali, Muhammad Shabir, Munazza Naz, Florentin Smarandache, Soft neutrosophic semigroups and their generalization, In Neutrosophic Theory and Its Applications.

Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 349367.

[263]

016

Neutrosophic Science International Association A. A. Salama, Said Broumi and Florentin Smarandache, Some Types of Neutrosophic Crisp Sets and Neutrosophic Crisp Relations, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 379-385.

[275] A. A. Salama, Florentin Smarandache, S. A. Alblowi, The Characteristic Function of a Neutrosophic Set, In Neutrosophic Sets and Systems, Vol. 3, 2014, pp. 14-17.

[276] Florentin Smarandache, Stefan Vladutescu, The Fifth Function of University: “Neutrosophic E-function” of CommunicationCollaboration-Integration of University in the Information Age, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 445-462.

[277] Vasile Patrascu, The Neutrosophic Entropy and its Five Components, In Neutrosophic Sets and Systems, Vol. 7, 2015, pp. 4046.

[278] Florentin Smarandache, Thesis-Antithesis-Neutrothesis, and Neutrosynthesis, In Neutrosophic Sets and Systems, Vol. 8, 2015, pp.

57-58.

[279] Florentin Smarandache, (t, i, f)-Neutrosophic Structures & INeutrosophic Structures (Revisited), In Neutrosophic Sets and Systems, Vol. 8, 2015, pp. 3-9.

[280] Florentin Smarandache, Sukanto Bhattacharya, To be and Not to be – An introduction to Neutrosophy: A Novel Decision Paradigm, In Neutrosophic Theory and Its Applications. Collected Papers, Volume 1, EuropaNova, Bruxelles, 2014, pp. 424-439.

[281] Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, TOPSIS method for multi-attribute group decision-making under singlevalued neutrosophic environment, In Neural Comput & Applic., 2015, 11 p.

[282] Pabitra Kumar Maji, Weighted Neutrosophic Soft Sets. In Neutrosophic Sets and Systems, Vol. 6, 2014, pp. 6-11.

[283] Pabitra Kumar Maji, Weighted Neutrosophic Soft Sets Approach in a Multicriteria Decision Making Problem. In Journal of New Theory, 2015, 12 p [274]

017

Neutrosophic Science International Association

‫ مقدمات لمؤتمرات عالمية وحلقات دراسية حول النيوتروسوفيك‬5.6.2 Presentations to International Conferences or Seminars [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

018

F. Smarandache, Foundations of Neutrosophic set and Logic and Their Applications to Information Fusion, Okayama University of Science, Kroumov Laboratory, Department of Intelligence Engineering, Okayama, Japan, 17 December 2013.

Jean Dezert & Florentin Smarandache, Advances and Applications of Dezert-Smarandache Theory (DSmT) for Information Fusion, presented by F. Smarandache, Osaka University, Department of Engineering Science, Inuiguchi Laboratory, Japan, 10 January 2014.

F. Smarandache, Foundations of Neutrosophic Set and Logic and Their Applications to Information Fusion, Osaka University, Inuiguchi Laboratory, Department of Engineering Science, Osaka, Japan, 10 January 2014.

F. Smarandache, Alpha-Discounting Method for Multicriteria Decision Making, Osaka University, Department of Engineering Science, Inuiguchi Laboratory, Japan, 10 January 2014.

F. Smarandache, The Neutrosophic Triplet Group and its Application to Physics, seminar Universidad Nacional de Quilmes, Department of Science and Technology, Buenos Aires, Argentina, 02 June 2014.

F. Smarandache, Foundations of Neutrosophic Logic and Set and their Applications to Information Fusion, tutorial, 17th International Conference on Information Fusion, Salamanca, Spain, 7th July 2014.

Said Broumi, Florentin Smarandache, New Distance and Similarity Measures of Interval Neutrosophic Sets, 17th International Conference on Information Fusion, Salamanca, Spain, 7-10 July 2014.

F. Smarandache, Foundations of Neutrosophic Logic and Set Theory and their Applications in Science. Neutrosophic Statistics and Neutrosophic Probability. n-Valued Refined Neutrosophic Logic, Universidad Complutense de Madrid, Facultad de Ciencia

Neutrosophic Science International Association Matemáticas, Departamento de Geometría y Topología, Instituto Matemático Interdisciplinar (IMI), Madrid, Spain, 9th July 2014.

[9] F. Smarandache, (T, I, F)-Neutrosophic Structures, Annual Symposium of the Institute of Solid Mechanics, SISOM 2015, Robotics and Mechatronics. Special Session and Work Shop on VIPRO Platform and RABOR Rescue Robots, Romanian Academy, Bucharest, 21-22 May 2015.

[10] Mumtaz Ali & Florentin Smarandache, Neutrosophic Soluble Groups, Neutrosophic Nilpotent Groups and Their Properties, Annual Symposium of the Institute of Solid Mechanics, SISOM 2015, Robotics and Mechatronics. Special Session and Work Shop on VIPRO Platform and RABOR Rescue Robots, Romanian Academy, Bucharest, 21-22 May 2015.

[11] V. Vladareanu, O. I. Sandru, Mihnea Moisescu, F. Smarandache, Hongnian Yu, Modelling and Classification of a Robotic Workspace using Extenics Norms, Annual Symposium of the Institute of Solid Mechanics, Robotics and Mechatronics. Special Session and Work Shop on VIPRO Platform and RABOR Rescue Robots, Romanian Academy, Bucharest, 21-22 May 2015.

[12] Luige Vladareanu, Octavian Melinte, Liviu Ciupitu, Florentin Smarandache, Mumtaz Ali and Hongbo Wang, NAO robot integration in the virtual platform VIPRO, Annual Symposium of the Institute of Solid Mechanics, SISOM 2015, Robotics and Mechatronics. Special Session and Work Shop on VIPRO Platform and RABOR Rescue Robots, Romanian Academy, Bucharest, 21-22 May 2015.

[13] F. Smarandache, Types of Neutrosophic Graphs and neutrosophic Algebraic Structures together with their Applications in Technology, Universitatea Transilvania din Brasov, Facultatea de Design de Produs si Mediu, Brasov, Romania, 06 June 2015.

‫ اطاريح دكتوراه في النيوتروسوفيا‬5.6.2 Ph. D. Dissertations [1]

019

Eng. Stefan Adrian Dumitru, Contributii in dezvoltarea sistemelor de control neuronal al miscarii robotilor mobili autonomi, adviser

Neutrosophic Science International Association

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

001

Dr. Luige Vlădăreanu, Institute of Solid Mechanics, Romanian Academy, Bucharest, 25 September, 2014.

Eng. Dănuț Adrian Bucur, Contribuţii în controlul mișcării sistemelor de prehensiune pentru roboți și mâini umanoide inteligente, adviser Dr. Luige Vlădăreanu, Institute of Solid Mechanics, Romanian Academy, Bucharest, 25 September, 2014.

Eng. Daniel Octavian Melinte, Cercetari teoretice si experimentale privind controlul sistemelor mecanice de pozitionare cu precizie ridicata, advisers Dr. Luige Vlădăreanu & Dr. Florentin Smarandache, Institute of Solid Mechanics, Romanian Academy, Bucharest, September 2014 .

Eng. Ionel Alexandru Gal, Contributions to the Development of Hybrid Force-Position Control Strategies for Mobile Robots Control, advisers Dr. Luige Vlădăreanu & Dr. Florentin Smarandache, Institute of Solid Mechanics, Romanian Academy, Bucharest, October 14, 2013.

Smita Rajpal, Intelligent Searching Techniques to Answer Queries in RDBMS, Ph D Dissertation in progress, under the supervision of Prof. M. N. Doja, Department of Computer Engineering Faculty of Engineering, Jamia Millia Islamia, New Delhi, India, 2011.

Josué Antonio Nescolarde Selva, A Systematic Vision of Belief Systems and Ideologies, under the supervision of Dr. Josep Llus Usó I Domènech, Dr. Francesco Eves Macià, Universidad de Alicante, Spain, 2010.

Ming Zhang, Novel Approaches to Image Segmentation Based on Neutrosophic Logic, Ph D Dissertation, Utah State University, Logan, Utah, USA, All Graduate Theses and Dissertations, Paper 795, 12-1-201, 2010.

Haibin Wang, Study on Interval Neutrosophic Set and Logic, Georgia State University, Atlanta, USA, 2005.

Sukanto Bhattacharya, Utility, Rationality and Beyond - From Finance to Informational Finance [using Neutrosophic Probability], Bond University, Queensland, Australia, 2004.

‫‪Neutrosophic Science International Association‬‬

‫إ اُزؾِ‪ َ٤‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٣ ٢‬ؼم رؼٔ‪ٔ٤‬ب ُزؾِ‪ َ٤‬أُغٔ‪ٞ‬ػخ ‪ٝ ،‬اُز‪٢‬‬ ‫ثم‪ٛ ٝ‬ب رؼم رؼٔ‪ٔ٤‬ب ُزؾِ‪ َ٤‬اُلز ح ‪ ،‬إ اُؾضجبٕ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬اُزٔ‪٤ٜ‬م‪١‬‬ ‫‪٣‬ر‪ ٤‬اُ‪ ٠‬ه‪ ْ٤‬صبثزخ ُالرؼ‪ ، ٖ٤٤‬ث‪ٔ٘٤‬ب ؽضجبٕ اُزلبظَ ‪ٝ‬اُزٌبَٓ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٌٖٔ٣ ٢‬اػزجب ‪ٛ‬ب ‪٣‬بظ‪٤‬بد ذاد الرؼ‪ٓ ٖ٤٤‬زـ‪، ٤‬‬ ‫ٓجب ئ ؽضبة اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ٝ ٢‬ؽضبة اُزلبظَ‬ ‫‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٌٖٔ٣ ٢‬رط‪ٔٛ ٣ٞ‬ب ثط م ػم‪٣‬مح ‪ٝ ،‬ذُي ثبالػزٔب‬ ‫ػِ‪ ٠‬اٗ‪ٞ‬اع اُالرؼ‪ ٖ٤٤‬اُز‪ ٢‬رٌِٔ‪ٜ‬ب أُض ُخ ًاُي ثبالػزٔب ػِ‪ ٠‬اُط م‬ ‫أُضزخمٓخ ك‪ ٢‬اُزؼبَٓ ٓؼ‪ٜ‬ب ‪.‬‬ ‫همٓ٘ب ك‪ٛ ٢‬اا اٌُزبة ثؼط االٓضِخ ػٖ اُالرؼ‪٘٤٤‬بد اُخبصخ ‪ٌُٖ ،‬‬ ‫‪ٞ٣‬عم اٌُض‪ ٖٓ ٤‬اُالرؼ‪٘٤٤‬بد ك‪ ٢‬ؽ‪٤‬بر٘ب اُ‪٤ٓٞ٤‬خ ‪٣ ،‬غت ػِ‪٘٤‬ب اصز‪ٜ‬ب ‪ٝ‬اػب ح‬ ‫ؽِ‪ٜ‬ب ثبصزخماّ غ م ٓربث‪ٜ‬خ ا‪ ٝ‬غ م ٓخزِلخ ػٖ اُز‪ ٢‬ذً ٗب‪ٛ‬ب ‪ُ .‬اُي ٗؾٖ‬ ‫ثؾبعخ اُ‪ٓ ٠‬ز‪٣‬م ٖٓ االثؾبس اُؼِٔ‪٤‬خ ك‪ ٢‬ؽوَ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كي ‪.‬‬ ‫‪ٓ ٌٖٔ٣‬ضزوجال اصخ ؽضجبٕ اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٢‬‬ ‫اُغزئ‪ ٖٓ ٢‬خالٍ اثؾبس ٓزومٓخ ك‪ٛ ٢‬اا أُعٔب ‪.‬‬ ‫همّ أُؤُق ك‪ٛ ٢‬اا اٌُزبة ‪ٝ‬ال‪ ٓ ٍٝ‬ح اكٌب ٌَُ ٖٓ شج‪ ٚ‬اُـب‪٣‬خ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ‪ٝ ،‬شجخ االصزٔ ا ‪٣‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ ثط م ٓخزِلخ ػٖ شج‪ٚ‬‬ ‫– االصزٔ ا ‪٣‬خ اُزوِ‪٤‬م‪٣‬خ‪ٝ ،‬شج‪ ٚ‬أُرزوخ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪٤‬خ‪ٝ ،‬شج‪ ٚ‬اُزٌبَٓ‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬كٌ‪ ٢‬ثط م رخزِق ػٖ ؽضجبٕ اُزلبظَ ‪ٝ‬اُزٌبَٓ اُغزئ‪٢‬‬ ‫االػز‪٤‬ب ‪ ، ١‬ثبالظبكخ اُ‪ ٠‬اُزؼب ‪٣‬ق اُزوِ‪٤‬م‪٣‬خ ُِـب‪٣‬خ ‪ٝ‬االصزٔ ا ‪٣‬خ ‪ٝ‬أُرزوخ‬ ‫‪ٝ‬اُزٌبَٓ رجبػب‪.‬‬

‫‪000‬‬

‫‪Neutrosophic Science International Association‬‬

‫ٗجاح ػٖ أُز عٔ‪ٖ٤٤‬‬ ‫‪ٛ .‬م‪ ٟ‬اصٔبػ‪ َ٤‬خبُم اُغٔ‪٢ِ٤‬‬ ‫‪ُٝ‬مد ػبّ ‪ 1314‬ك‪ٓ ٢‬ؾبكظخ ٗ‪ / ٟٞ٘٤‬اُؼ ام ‪.‬ؽصعععععِذ ػِ‪ ٠‬ش‪ٜ‬عععب ح اُجٌبُعععع‪ٞ٣ ٞ‬س ك‪ ٢‬ػِعععع‪ّٞ‬‬ ‫اُ ‪٣‬بظ‪٤‬بد ٖٓ هضْ اُ ‪٣‬بظ‪٤‬بد ‪٤ًِ /‬خ اُؼِ‪ /ّٞ‬عبٓؼخ أُ‪ٞ‬صَ ‪ٝ‬ؽصِععذ ػِ‪ ٠‬ش‪ٜ‬ب ح أُععبعضز‪ٖٓ ٤‬‬ ‫ٗلش اُغعبٓؼخ ٖٓ ًِ‪٤‬خ ػِ‪ ّٞ‬اُؾبص‪ٞ‬ة ‪ٝ‬اُ ‪٣‬بظ‪٤‬بد‪ /‬هضْ اُ ‪٣‬بظ‪٤‬بد ػٖ صبُز‪ٜ‬ب أُ‪ٞ‬ص‪ٓٞ‬خ‬ ‫"اصعععععزخماّ اُـعععالف اُطعععع‪٤‬ل‪ ٢‬ك‪ٞٓ ٢‬از‪٣‬عععععٖ اُرععععؼ اُؼ ثعععع‪ " ٢‬ػععبّ ‪ًٔ ،1221‬ب ‪ٝ‬ؽصععِذ‬ ‫ػِ‪ ٠‬شعع‪ٜ‬ب ح اُمًز‪ ٞ‬ا‪ ٙ‬ك‪ ٢‬اُ ‪٣‬بظ‪٤‬بد ٖٓ ٗلش اٌُِ‪٤‬خ ػٖ وغ ‪ٝ‬ؽز‪ٜ‬ب أُ‪ٞ‬ص‪ٓٞ‬خ "اُزوص‪ ٢‬ك‪ ٢‬اُزؾِ‪َ٤‬‬ ‫اُؾضععبس ك‪ٓ ٢‬ضبئععَ اُج ٓغخ اُ‪ٜ٘‬مص‪٤‬خ أُؼٔ​ٔخ "‪ُ .‬م‪ٜ٣‬ب ػع‪٣ٞ‬خ ك‪ ٢‬اًب ‪٤ٔ٣‬خ ‪Telesio- Galilei‬‬ ‫اُؼِٔ‪٤‬خ اُِ٘مٗ‪٤‬خ ‪ٝ ،‬ػع‪ٞ‬ح ك‪٤ٛ ٢‬ئخ رؾ ‪ ٣‬أُغِخ اُؼبُٔ‪٤‬خ ‪ ٝ ، JHEPGC‬ػع‪ٞ‬ح ش ف ك‪ ٢‬أُغٔغ اُؼِٔ‪٢‬‬ ‫اُؼبُٔ‪ ٢‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٘ٓ ٢‬ا ‪ٝ ، 1211/1/13‬ػع‪ٞ‬ح ك‪٤ٛ ٢‬ئخ رؾ ‪ٓ ٣‬غِخ ‪ NSS‬اُزبثؼخ ُغبٓؼخ ٗ‪ٌٓ ٞ٤‬ض‪ٌٞ٤‬‬ ‫االٓ ‪٤ٌ٣‬خ‪ًٔ ،‬ب ‪ًٝ‬بٗذ ئ‪٤‬ضخ ُوضْ اُ ‪٣‬بظ‪٤‬بد ك‪٤ًِ ٢‬خ اُز ث‪٤‬خ االصبص‪٤‬خ ‪ /‬عبٓؼخ أُ‪ٞ‬صَ‪-‬عبٓؼخ رِؼل ٖٓ‬ ‫‪ 1211‬اُ‪.1211 ٠‬ؽصِذ ػِ‪ ٠‬ش‪ٜ‬ب ح روم‪٣ ٣‬خ ٖٓ ٓؼبُ‪ٝ ٢‬ز‪ ٣‬اُزؼِ‪ ْ٤‬اُؼبُ‪ٝ ٢‬اُجؾش اُؼِٔ‪ ٢‬اُؼ اه‪ُٔ ٢‬رب ًز‪ٜ‬ب‬ ‫ك‪ ٢‬عبئزح اُزؼِ‪ ْ٤‬اُؼبُ‪ُِ ٢‬ؼِ‪ ّٞ‬ػبّ ‪ُٜ .1211‬ب ا‪ٛ‬زٔبٓبد ثؾض‪٤‬خ ػم‪٣‬مح ا‪ٜٔٛ‬ب أُ٘طن اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ ، ٢‬اُج ٓغخ‬ ‫اُ‪ٜ٘‬مص‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ‪ ،‬أُؼب الد اُؼاله‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ أُعججخ ‪ ،‬اُج ٓغخ اُ‪ٜ٘‬مص‪٤‬خ أُعججخ ‪ ،‬اُج ٓغخ‬ ‫اُ‪ٜ٘‬مص‪٤‬خ أُؼٔ​ٔخ ‪،‬رطج‪٤‬وبد اُج ٓغخ اُ‪ٜ٘‬مص‪٤‬خ‪ .‬اؽمس ٓ٘ر‪ ٞ‬ار‪ٜ‬ب ك‪ٓ ٢‬غبٍ رخصص‪ٜ‬ب ‪ٝ‬أُ٘طن اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٢‬‬ ‫ًبٕ ؽ‪ٝ ٍٞ‬ظغ ص‪٤‬ؾ عم‪٣‬مح ‪ٓٝ‬جزٌ ح ُِؾِ‪ ٍٞ‬اُؼظٔ‪ٝ ٠‬اُمٗ‪٤‬ب ك‪ ٢‬أُؼب الد اُؼاله‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ‪ً ،‬اُي ‪ٝ‬ظغ‬ ‫ٓل‪ٓ ّٜٞ‬جزٌ ٍ (والهَ ا‪٣ ٝ‬ضب‪ )١ٝ‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ ٢‬ك‪ٓ ٢‬ضبئَ اُج ٓغخ اُ‪ٜ٘‬مص‪٤‬خ ؿ‪ ٤‬أُو‪٤‬مح ‪ُٜٝ .‬ب ثؾ‪ٞ‬س اخ ‪ٟ‬‬ ‫‪ًٝ‬زت ه‪٤‬م اُطجبػخ ‪ٝ‬اُ٘ر ُِٔز‪٣‬م ‪ ٌٖٔ٣‬اُ ع‪ٞ‬ع اُ‪ ٠‬أُ‪ٞ‬ص‪ٞ‬ػخ اُؼبُٔ‪٤‬خ ُِجبؽض‪ ٖ٤‬ك‪ ٢‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ي (‪. (ENR‬‬

‫أُ‪ٜ٘‬مس اٌُ‪ ٜ‬ثبئ‪ : ٢‬وؽٔم خع ػ‪٤‬ض‪ ٠‬اُغج‪١ ٞ‬‬ ‫‪ُٝ‬م ػبّ ‪ 1321‬ك‪ٓ ٢‬ؾبكظخ ٗ‪ / ٟٞ٘٤‬اُؼ ام ‪ .‬ؽصَ ػِ‪ ٠‬ش‪ٜ‬ب ح اُجٌبُ‪ٞ٣ ٞ‬س ك‪٘ٛ ٢‬مصخ اُوم ح‬ ‫اٌُ‪ ٜ‬ثبئ‪٤‬خ ‪ /‬اٌُِ‪٤‬خ اُزو٘‪٤‬خ اُ‪ٜ٘‬مص‪٤‬خ ثبُٔ‪ٞ‬صَ ػبّ (‪٣ )1222-1221‬ؼَٔ ك‪ ٢‬عبٓؼخ رِؼل ‪ً /‬عِ‪٤‬خ‬ ‫اُز ث‪٤‬خ االصبص‪٤‬خ ٓ٘ا ػبّ ‪ُٝ 1211‬ؾم االٕ‪ .‬ثبالظبكخ اُ‪ ٠‬ا‪ٛ‬زٔبٓ‪ٜ‬خ ك‪ٓ ٢‬غبٍ اخزصبص‪ ، ٚ‬ك‪ٜٞ‬‬ ‫ٓ‪ٜ‬زْ ا‪٣‬عب ثبُ ‪٣‬بظ‪٤‬بد ‪ٝ‬اُل‪٤‬ز‪٣‬بء خصعع‪ٞ‬صب ك‪ٔ٤‬ب ‪٣‬زؼِن ك‪ ٢‬أُ٘طعععن اُ٘‪ٞ٤‬ر ‪ٝ‬صععع‪ٞ‬ك‪ٝ ٢‬أُ٘طن‬ ‫اُعجبث‪ٝ ٢‬ػِْ اٌُ‪٤ٗٞ‬بد‪ .‬ؽصَ ك‪ 1216/4/16 ٢‬ػِ‪ ٠‬ػع‪٣ٞ‬خ ش ف أُغٔغ اُؼِععٔ‪ ٢‬اُؼعبُٔ‪٢‬‬ ‫اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ُٝ ، ٢‬م‪ ٚ٣‬اكٌب ٓجزٌ ح ُزط‪ ٣ٞ‬اال ‪ٝ‬اد اُ ‪٣‬بظعع‪٤‬خ ‪٣ٝ‬ؼَٔ ػِ‪ ٠‬ث٘بء ٗععظ ‪٣‬خ اُزعوبثَ‬ ‫ك‪ ٢‬اُج ٓغخ اُ‪ٜ٘‬مص‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ‪ًٔ ،‬ب هبّ ث‪ٞ‬ظغ ثؼط أُخططبد اُزج‪ُٞٞ‬ع‪٤‬خ ُِلعبء أُز ‪ ١‬اُغزئ‪ ٢‬ك‪٢‬‬ ‫ؽضجبٕ اُزلبظَ ‪ٝ‬اُزٌبَٓ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ ٖٓٝ ، ٢‬ظٖٔ ا‪ٛ‬زٔبٓبر‪ ٚ‬اُجؾض‪٤‬خ اُج ٓغخ اُ‪ٜ٘‬مص‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ‪،‬‬ ‫أُؼب الد اُؼاله‪٤‬خ اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬خ ‪ ،‬أُؼب الد اُؼاله‪٤‬خ أُعججخ‪ ،‬اُج ٓغخ اُ‪ٜ٘‬مص‪٤‬خ أُعججخ ‪ ،‬اُج ٓغخ اُ‪ٜ٘‬مص‪٤‬خ‬ ‫‪،‬رطج‪٤‬وبد اُج ٓغخ اُ‪ٜ٘‬مص‪٤‬خ ك‪٘ٛ ٢‬مصخ اُوم ح اٌُ‪ ٜ‬ثبئ‪٤‬خ ‪ ُٚٝ‬اثؾبس ؽم‪٣‬ضخ ٓ٘ر‪ ٞ‬ح ك‪ٓ ٢‬غالد ػبُٔ‪٤‬خ ٓخزصخ‬ ‫ثبُٔ٘طن اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪ٓ ُٚٝ ٢‬ؤُلبد ه‪٤‬م اُطجبػخ ‪ٝ‬اُ٘ر ‪ُِٔٝ‬ز‪٣‬م ‪ ٌٖٔ٣‬اُ ع‪ٞ‬ع اُ‪ ٠‬أُ‪ٞ‬ص‪ٞ‬ػخ اُؼبُٔ‪٤‬خ ُِجبؽض‪ٖ٤‬‬ ‫ك‪ ٢‬اُ٘‪ٞ٤‬ر ‪ٝ‬ص‪ٞ‬ك‪٤‬ي )‪.)ENR‬‬

‫‪002‬‬

‫‪Neutrosophic Science International Association‬‬

‫فرَك انؼًم‬ ‫أُؤُق ‪ :‬كِ‪ٗ ٞ‬زٖ صٔ اٗماًخ‬

‫آ ‪ٌ٣‬ب عبٓؼخ ٗ‪ٌٓ ٞ٤‬ض‪ٌٞ٤‬‬

‫أُز عٔ‪ٛ : ٕٞ‬م‪ ٟ‬اُغٔ‪ – ٢ِ٤‬اؽٔم اُغج‪ ١ ٞ‬اُؼ ام عبٓؼخ رِؼل‬ ‫أُ اعؼ‪ : ٕٞ‬اؽٔم صالٓخ – ‪٣ٞٛ‬ما اُـ‪ٞ‬اُج‪٢‬‬ ‫صؼ‪٤‬م ث ‪٢ٓٝ‬‬ ‫أُ اعغ اُِـ‪ : ١ٞ‬ػجمأُ٘ؼْ اُمُ‪٢ٔ٤‬‬

‫ٓص‬

‫عبٓؼخ ث‪ ٞ‬صؼ‪٤‬م‬

‫أُـ ة اُؼ ث‪٢‬‬ ‫اُؼ ام‬

‫عبٓؼخ رِؼل‬

‫‪003‬‬

Neutrosophic Science International Association

Neutrosophic Precalculus and Neutrosophic Calculus Authorship Florentin Smarandache University of New Mexico, 705 Gurley Ave. Gallup, NM 87301, USA.

smarand@unm.edu

Transolators Huda E. Khalid1 Ahmed K. Essa2 1University of Telafer, Department of Mathematics, College of Basic Education, Mosul, Iraq. hodaesmail@yahoo.com 2 University of Telafer, College of Basic Education, Mosul, Iraq. ahmed.ahhu@gmail.com Peer Reviewers:

A.A. Salama1, Hewayda ElGhawalby2 Said Broumi3 Abdelmoneim A. Al-Dulaimi4 1Department

of Mathematics and Computer Science, Faculty of Science, Port Said University, Egypt 2Physics and Engineering Mathematics Department,Faculty of Engineering, Port Said University, Egypt 3 Laboratory of Information Processing, Faculty of Science Ben M'Sik. University Hassan II , B.P 7055 sidiothman , Casablanca Morocco broumisaid78@gmail.com 4 Department of Arabic Languge, College of Basic Education, University of Telafer , Mosul , Iraq.

E-mail: abdmo469@yahoo.com

004