Neutrosophic Science International Association
يثادئ انرفاضم وانركايم انُُىذروصىفكً و حضاب انرفاضم وانركايم انُُىذروصىفكً 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 انُُرروصىفكُح
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 ٢غ٘٘ب 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 انُُىذروصىفكٍ 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 انُُىذروصىفكٍ 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 انُُىذروصىفكٍ 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
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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
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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 انُُىذروصىفٍ) ،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 ٣خٝٝ ،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.
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Neutrosophic Science International Association
انفصم االول انرًهُذ (انًفهىو انُُىذروصىفكٍ 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 انُُىذروصىفكٍ 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
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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.
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ٝػِ ٠ذُي كبٕ رؾِ َ٤أُغٔٞػخ ٣ؼم رؼٔٔ٤ب ُزؾِ َ٤اُلز ح. انرحهُم انُُىذروصىفكٍ (او انحضاب انُُىذروصىفكٍ ) ٌٖٔ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 انُُىذروصىفكٍ ٓ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.
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٘ٛ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.
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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}.
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g1: R2
g2: R
f3: Rرخزِق كٓ ٢غبُٜب ٓٝغبُ ٚأُوبثَ ؽ٤ش
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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
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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 انًؼُُح 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 أُٔبس ،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
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Neutrosophic Science International Association
– 5.1لىاٍَُ فُزَائُح غُر يؼُُح 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 رٌٔ٘٘ب ٖٓ 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.
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Neutrosophic Science International Association
انفصم انصاٍَ يثادئ انرفاضم وانركايم انُُىذروصىفكٍ - 1.2انؼًهُاخ انجثرَح نهًجايُغ ُزٌٖ )(21 )(22 )(23 )(24 )(25
ٓ ٝغٔٞػزبٕ
∈
ٞٛو ١ه٤بص ٢ػ٘مئا {
;} ∈ | { ∈ | ;} ∈ { ∈ | ;} ∈ ∈ | { ;} ∈ -
∈
∈ | ,
-2.2انؼاللاخ تٍُ انًجايُغ انجزئُح انُُىذروصىفكُح ُ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 ٢اٌُُ٘٘ 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 ).
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Neutrosophic Science International Association
-3.2دانح انًجًىػح انجزئُح انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 الٌ٘٘ٔ٣ٝ .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
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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 .
-2.2انالذؼٍُُ (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 رٌٔ٘٘ب ٖٓ 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.
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Neutrosophic Science International Association
-2.2انذوال انضًُُح انُُىذروصىفكُح ٝ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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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
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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
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Neutrosophic Science International Association
- 12.2أصفار انذانح انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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.
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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 انُُىذروصىفكٍ 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
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ص٘ؼَٔ ٓغ ٓغٔٞػخ ث٤بٗبد رو ٣ج٤خ ك ٢االٗؾما اُ٘ٞ٤ر ٝصٞكٌ: ٢ ) )(63
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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 انُُىذروصىفكٍ ٓ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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفٍ -1.3انغاَح انُُىذروصىفكُح 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 |{ |
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-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
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|{ }| || ٝ
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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفكٍ 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انمًًُ انىصطً انُُىذروصىفكُح ُ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 أُؼٔٔخ ُٜ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 انمًًُ انىصطً انُُىذروصىفكُح ُ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 أُظِِخ ٝك ٢ؽو٤وخ االٓ ٓضَ . لٕ ٛا ٙأُ٘طوخ رٔضَ صٔب ُِماُخ اُ٘ٞ٤ر ٝصٞكٌ٤خ ك ٢اُلز ح
-14.3يصال حىل َظرَح انمًًُ انىصطً انُُىذروصىفكُح 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 انُُىذروصىفكُح ٝ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 }
}
{ {
}
{
}
}
{(
{
{
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)(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 {
}
{
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اٝ }
)(154
{
{
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{
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اٝ )} )(155
{
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)}
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{( {
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{ 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح 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 انُُىذروصىفكُح ُ٘ -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 انُُىذروصىفكٍ 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 انُُىذروصىفكٍ 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 انُُىذروصىفٍ 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 أُظِِخ) ،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 انُُىذروصىفكٍ 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 انُُىذروصىفكٍ 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.
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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.
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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
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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
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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
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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
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ٕٞٗ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
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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
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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]
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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]
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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]
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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.
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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]
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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.
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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]
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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]
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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]
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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.
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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Neutrosophic Science International Association
مقدمات لمؤتمرات عالمية وحلقات دراسية حول النيوتروسوفيك5.6.2 Presentations to International Conferences or Seminars [1]
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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]
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Eng. Stefan Adrian Dumitru, Contributii in dezvoltarea sistemelor de control neuronal al miscarii robotilor mobili autonomi, adviser
Neutrosophic Science International Association
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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.
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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 أُؼٔٔخ "ُ .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 أُؼٔٔخ ،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
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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
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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
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