FUZZY NEUTROSOPHIC RELATIONS

[Arockiarani et. al., Vol.4 (Iss.2): February, 2016]

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FUZZY NEUTROSOPHIC RELATIONS I. Arockiarani 1, J. Martina Jency 2 1 Department of Mathematics, Nirmala College for women, Coimbatore, Tamilnadu, INDIA 2 Department of Mathematics, Coimbatore Institute of Engineering and Technology, Coimbatore, Tamilnadu, INDIA ABSTRACT The focus of this paper is to present the concept of fuzzy neutrosophic relations. Further we study the composition of fuzzy neutrosophic relations with the choice of t-norms and tconorms and characterize their properties.

Keywords: Fuzzy neutrosophic set, fuzzy neutrosophic relation.

Cite This Article: I. Arockiarani, J. Martina Jency, “FUZZY NEUTROSOPHIC RELATIONS” International Journal of Research – Granthaalayah, Vol. 4, No. 2 (2016): 17-30.

1. INTRODUCTION A relation is a mathematical description of a situation where certain elements of sets are related to one another in some way. It is a tool for describing correspondences between objects. The use of fuzzy relations originated from the observation that real life objects can be related to each other to certain degree. Fuzzy relations are able to model vagueness, but they cannot model uncertainty .Intuitionistic fuzzy sets, as defined by Atanassov [4], give us a way to incorporate uncertainty in an additional degree. In 1995, F.Smarandache [13, 14] combined the non-standard analysis with a tri component logic/set, probability theory and with philosophy and proposed the term neutrosophic which means knowledge of neutral thoughts. This neutral represents the main distinction between fuzzy and intuitionistic fuzzy logic set. Motivated by the concept which deals with non-standard analysis I.Arockiarani et al. [1, 2] defined the fuzzy neutrosophic set involving the concept of standard analysis. In this paper we study the properties of fuzzy neutrosophic relations in a set and the properties of the composition with different t-norms and tconorms.

2. PRELIMINARIES In order to define the fuzzy neutrosophic relations, we will use the well –known triangular norms and conorms in [0, 1], taking into account that as non-classical connectives. They do not satisfy

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the boolean standard identities. We will call t-norm in [0, 1] to every mapping [0,1] ď‚´ [0, 1] ď‚Ž [0, 1] satisfying the following properties 1.

2.

3.

4.

T:

Boundary conditions, T(x, 1) = x and T(x, 0) = 0, for all x ďƒŽ [0, 1] Monotony, T(x, y) ≤ T (z, t) if x ≤ z and y ≤ t Commutative, T(x, y) = T(y, x), for all x, y ďƒŽ [0, 1] Associative, T(T(x, y), z) = T(x, T(y, z)), for all x, y, z ďƒŽ [0, 1]

Given a t-norm T, we can consider the mapping S: [0,1] ď‚´ [0, 1] ď‚Ž [0, 1] S(x, y) ď‚ş1-T(1-x, 1-y).

This mapping S, will be called dual t-conorm of T. The most important properties of t-norms and t-conorms can be found in [9,11].Here we present the following theorem with regard to the distributive property of t-norms and t-conorms. In this paper unless it is said in the opposite way, we will designate the t-norms and t-conorms with the greek letters đ?›ź, đ?›˝, đ?œ†, đ?œŒ. Let I be a finite family of indices and {đ?‘Žđ?‘– }đ?‘–∈đ??ź , {đ?‘?đ?‘– }đ?‘–∈đ??ź be number collection of [0, 1].For every đ?›ź t-norm or tconorm and for every đ?œ† t-norm or t-conorm 1. đ?›źđ?‘–(đ?‘Žđ?‘– ∨ đ?‘?đ?‘– ) ≼ đ?›źđ?‘–(đ?‘Žđ?‘– ) ∨ đ?›źđ?‘–(đ?‘?đ?‘– ) 2. đ?œ†đ?‘– (đ?‘Žđ?‘– ∧ đ?‘?đ?‘– ) ≤ đ?œ†đ?‘–(đ?‘Žđ?‘– ) ∧ đ?œ†đ?‘– (đ?‘?đ?‘– ) are verified.

With the result given by L.W.Fung and S.K.Ku [10] relative to the fact that đ?›ź is an idempotent tconorm (idempotent t-norm) if and only if đ?›ź =∨ (đ?›ź =∧), we get the following theorem: Theorem 2.0: Let {đ?‘Žđ?‘– }đ?‘–∈đ??ź , {đ?‘?đ?‘– }đ?‘–∈đ??ź be two finite number families of [0,1] and đ?›ź, đ?œ† t-norms or t-conorms not null.

Then đ?›ź đ?›ź đ?›ź đ?‘–(đ?‘Žđ?‘– ∨ đ?‘?đ?‘– ) = đ?‘– (đ?‘Žđ?‘– ) ∨ đ?‘– (đ?‘?đ?‘– ) if and only if đ?›ź =∨ đ?œ† đ?œ† đ?œ† đ?‘– (đ?‘Žđ?‘– ∧ đ?‘?đ?‘– ) = đ?‘–(đ?‘Žđ?‘– ) ∧ đ?‘–(đ?‘?đ?‘– ) if and only if đ?œ† =∧ .

Definition 2.1: [2] A Fuzzy neutrosophic set A on the universe of discourse X is defined as A= ⌊đ?‘Ľ, đ?‘‡đ??´ (đ?‘Ľ), đ??źđ??´ (đ?‘Ľ), đ??šđ??´ (đ?‘Ľ)âŒŞ, đ?‘Ľ ∈ đ?‘‹ where đ?‘‡, đ??ź, đ??š: đ?‘‹ →[0, 1] and 0 ď‚Ł TA ( x)  I A ( x)  FA ( x) ď‚Ł 3

3. FUZZY NEUTROSOPHIC RELATIONS Definition 3.1: A fuzzy neutrosophic set relation is defined as a fuzzy neutrosophic subset of đ?‘‹ Ă— đ?‘Œ having the form đ?‘… = { ⌊(đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ś), đ??źđ?‘… (đ?‘Ľ, đ?‘Ś), đ??šđ?‘… (đ?‘Ľ, đ?‘Ś)âŒŞ: đ?‘Ľ ∈ đ?‘‹, đ?‘Ś ∈ đ?‘Œ } where đ?‘‡đ?‘… , đ??źđ?‘… , đ??šđ?‘… : đ?‘‹ Ă— đ?‘Œ → [0,1] Satisfy the condition 0 ≤ đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ś) + đ??źđ?‘… (đ?‘Ľ, đ?‘Ś) + đ??šđ?‘… (đ?‘Ľ, đ?‘Ś) ≤ 3 ∀ (đ?‘Ľ, đ?‘Ś) ∈ đ?‘‹ Ă— đ?‘Œ.

We will denote with đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ) the set of all fuzzy neutrosophic subsets in đ?‘‹ Ă— đ?‘Œ.

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Definition 3.2: Given a binary fuzzy neutrosophic relation between 𝑋 and 𝑌, we can define 𝑅 −1 between 𝑌 and 𝑋 by means of 𝑇𝑅−1 (𝑦, 𝑥) = 𝑇𝑅 (𝑥, 𝑦), 𝐼𝑅−1 (𝑦, 𝑥) = 𝐼𝑅 (𝑥, 𝑦), 𝐹𝑅−1 (𝑦, 𝑥) = 𝐹𝑅 (𝑥, 𝑦) ∀(𝑥, 𝑦) ∈ 𝑋 × 𝑌 to which we call inverse relation of R. Definition 3.3: Let 𝑅 and 𝑃 be two fuzzy neutrosophic relations between 𝑋 and 𝑌, for every (𝑥, 𝑦) ∈ 𝑋 × 𝑌 We can define, 1)𝑅 ≤ 𝑃 ⇔ 𝑇𝑅 (𝑥, 𝑦) ≤ 𝑇𝑃 (𝑥, 𝑦), 𝐼𝑅 (𝑥, 𝑦) ≤ 𝐼𝑃 (𝑥, 𝑦), 𝐹𝑅 (𝑥, 𝑦) ≥ 𝐹𝑃 (𝑥, 𝑦) 2)𝑅 ≼ 𝑃 ⇔ 𝑇𝑅 (𝑥, 𝑦) ≤ 𝑇𝑃 (𝑥, 𝑦), 𝐼𝑅 (𝑥, 𝑦) ≤ 𝐼𝑃 (𝑥, 𝑦), 𝐹𝑅 (𝑥, 𝑦) ≤ 𝐹𝑃 (𝑥, 𝑦) 3)𝑅 ∨ 𝑃 = { 〈(𝑥, 𝑦), 𝑇𝑅 (𝑥, 𝑦) ∨ 𝑇𝑃 (𝑥, 𝑦), 𝐼𝑅 (𝑥, 𝑦) ∨ 𝐼𝑃 (𝑥, 𝑦), 𝐹𝑅 (𝑥, 𝑦) ∧ 𝐹𝑃 (𝑥, 𝑦)〉 4)𝑅 ∧ 𝑃 = { 〈(𝑥, 𝑦), 𝑇𝑅 (𝑥, 𝑦) ∧ 𝑇𝑃 (𝑥, 𝑦), 𝐼𝑅 (𝑥, 𝑦) ∧ 𝐼𝑃 (𝑥, 𝑦), 𝐹𝑅 (𝑥, 𝑦) ∨ 𝐹𝑃 (𝑥, 𝑦)〉 5)𝑅 𝑐 = { 〈(𝑥, 𝑦), 𝐹𝑅 (𝑥, 𝑦), 1 − 𝐼𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑥, 𝑦)〉: 𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌} Theorem 3.4: Let R, P, Q be three elements of fuzzy neutrosophic relations (X × Y) (𝑖)𝑅 ≤ 𝑃 ⇒ 𝑅 −1 ≤ 𝑃−1 ( 𝑖𝑖)(𝑅 ∨ 𝑃)−1 = 𝑅 −1 ∨ 𝑃−1 (𝑖𝑖𝑖)(𝑅 ∧ 𝑃)−1 = 𝑅 −1 ∧ 𝑃−1 (𝑖𝑣)(𝑅−1 )−1 = 𝑅 (𝑣) 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∧ 𝑃) ∨ (𝑅 ∧ 𝑄), 𝑅 ∨ (𝑃 ∧ 𝑄) = (𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄) (𝑣𝑖)𝑅 ∨ 𝑃 ≥ 𝑅, 𝑅 ∨ 𝑃 ≥ 𝑃, 𝑅 ∧ 𝑃 ≤ 𝑅, 𝑅 ∧ 𝑃 ≤ 𝑃 (𝑣𝑖𝑖) 𝐼𝑓 𝑅 ≥ 𝑃 𝑎𝑛𝑑 𝑅 ≥ 𝑄, 𝑡ℎ𝑒𝑛 𝑅 ≥ 𝑃 ∨ 𝑄 (𝑣𝑖𝑖𝑖) 𝐼𝑓 𝑅 ≤ 𝑃 𝑎𝑛𝑑 𝑅 ≤ 𝑄, 𝑡ℎ𝑒𝑛 𝑅 ≤ 𝑃 ∨ 𝑄 Proof: (𝒊) 𝐼𝑓 𝑅 ≤ 𝑃 then 𝑇𝑅−1 (𝑦, 𝑥) = 𝑇𝑅 (𝑥, 𝑦) ≤ 𝑇𝑃 (𝑥, 𝑦) = 𝑇𝑃−1 (𝑦, 𝑥) 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦(𝑥, 𝑦) ∈ 𝑋 × 𝑌 𝐼𝑅−1 (𝑦, 𝑥) = 𝐼𝑅 (𝑥, 𝑦) ≤ 𝐼𝑃 (𝑥, 𝑦) = 𝐼𝑃−1 (𝑦, 𝑥) ∀(𝑥, 𝑦) ∈ 𝑋 × 𝑌 and 𝐹𝑅−1 (𝑦, 𝑥) = 𝐹𝑅 (𝑥, 𝑦) ≥ 𝐹𝑃 (𝑥, 𝑦) = 𝐹𝑃−1 (𝑦, 𝑥) ∀(𝑥, 𝑦) ∈ 𝑋 × 𝑌 .Hence 𝑅 −1 ≤ 𝑃−1 (𝒊𝒊) 𝑇(𝑅∨𝑃)−1 (𝑦, 𝑥) = 𝑇𝑅∨𝑃 (𝑥, 𝑦) = 𝑇𝑅 (𝑥, 𝑦) ∨ 𝑇𝑃 (𝑥, 𝑦) = 𝑇𝑅−1 (𝑦, 𝑥) ∨ 𝑇𝑃−1 (𝑦, 𝑥) = 𝑇𝑅−1 ∨𝑃−1 (𝑦, 𝑥).The proofs for 𝐼(𝑅∨𝑃)−1 (𝑦, 𝑥) = 𝐼𝑅−1 ∨𝑃−1 (𝑦, 𝑥) & 𝐹(𝑅∨𝑃)−1 (𝑦, 𝑥) = 𝐹𝑅−1 ∨𝑃−1 (𝑦, 𝑥) are done in a similar way.Hence (𝑅 ∨ 𝑃)−1 = 𝑅 −1 ∨ 𝑃−1 . (𝒊𝒊𝒊) Proof is similar to (ii) (𝒊𝒗) proof follows from the definition. (𝒗) We will use the fact that the operators ∨ and ∧ satisfy the distributive property when they are applied to elements of [0, 1].𝑇𝑅∧(𝑃∨𝑄) (𝑥, 𝑦) = 𝑇𝑅 (𝑥, 𝑦) ∧ {𝑇𝑃 (𝑥, 𝑦) ∨ 𝑇𝑄 (𝑥, 𝑦)} = { 𝑇𝑅 (𝑥, 𝑦) ∧ 𝑇𝑃 (𝑥, 𝑦)} ∨ { (𝑇𝑅 (𝑥, 𝑦) ∧ 𝑇𝑄 (𝑥, 𝑦)} = 𝑇𝑅∧𝑃 (𝑥, 𝑦) ∧ 𝑇𝑅∧𝑄 (𝑥, 𝑦) = 𝑇(𝑅∧𝑃) ∨ (𝑅∧𝑄) (𝑥, 𝑦) . Similarly 𝐼𝑅∧(𝑃∨𝑄) (𝑥, 𝑦) = 𝐼(𝑅∧𝑃) ∨ (𝑅∧𝑄) (𝑥, 𝑦) 𝑎nd 𝐹𝑅∧(𝑃∨𝑄) (𝑥, 𝑦) = 𝐹(𝑅∧𝑃) ∨ (𝑅∧𝑄) (𝑥, 𝑦). Hence 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∧ 𝑃) ∨ (𝑅 ∧ 𝑄).

The proof is analogous to the previous one, in the case of 𝑅 ∨ (𝑃 ∧ 𝑄) = (𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄). (𝒗𝒊)Proof is obvious. (𝒗𝒊𝒊)𝐼𝑓 𝑅 ≥ 𝑃 𝑎𝑛𝑑 𝑅 ≥ 𝑄 , 𝑇𝑅 ≥ 𝑇𝑃 𝑎𝑛𝑑 𝑇𝑅 ≥ 𝑇𝑄, 𝐼𝑅 ≥ 𝐼𝑃 𝑎𝑛𝑑 𝐼𝑅 ≥ 𝐼𝑄, 𝐹𝑅 ≤ 𝐹𝑃 𝑎𝑛𝑑 𝐹𝑅 ≤ 𝐹𝑄, ⇒ 𝑇𝑅 ≥ 𝑇𝑃 ∨ 𝑇𝑄, 𝐼𝑅 ≥ 𝐼𝑃 ∨ 𝐼𝑄 , 𝐹𝑅 ≤ 𝐹𝑃 ∧ 𝐹𝑄, ⇒ 𝑇𝑅 ≥ 𝑇𝑃∨𝑄 , 𝐼𝑅 ≥ 𝐼𝑃∨𝑄 , 𝐹𝑅 ≤ 𝐹𝑃∨𝑄 ⇒ 𝑅 ≥ 𝑃 ∨ 𝑄 Http://www.granthaalayah.com ©International Journal of Research - GRANTHAALAYAH

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Similarly we can prove (đ?’—đ?’Šđ?’Šđ?’Š) .We can generalize the operations between binary fuzzy neutrosophic relations R, Q ∈ FNR (X Ă— Y). Using the well known triangular t-norms and tconorms in [0, 1]. For a triangular t – norms T and its dual t – conforms S, we get đ?‘‡(đ?‘…, đ?‘„) = { ⌊(đ?‘Ľ, đ?‘Ś), đ?‘‡(đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡(đ??źđ?‘… (đ?‘Ľ, đ?‘Ś), đ??źđ?‘„ (đ?‘Ľ, đ?‘Ś)), đ?‘†(đ??šđ?‘… (đ?‘Ľ, đ?‘Ś), đ??šđ?‘„ (đ?‘Ľ, đ?‘Ś))âŒŞ } đ?‘†(đ?‘…, đ?‘„) = { ⌊(đ?‘Ľ, đ?‘Ś), đ?‘†(đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘†(đ??źđ?‘… (đ?‘Ľ, đ?‘Ś), đ??źđ?‘„ (đ?‘Ľ, đ?‘Ś)), đ?‘‡(đ??šđ?‘… (đ?‘Ľ, đ?‘Ś), đ??šđ?‘„ (đ?‘Ľ, đ?‘Ś))âŒŞ } COMPOSITION OF FUZZY NEUTROSOPHIC RELATIONS Basing ourselves on the composition of binary IF relations in [0, 1] we can give the following definitions.

Definition 3.5: Let đ?›ź, đ?›˝, đ?œ†, đ?œŒ be t-norms or t-conorms not necessarily dual two – two, and đ?‘ƒ ∈ đ??šđ?‘ đ?‘…(đ?‘Œ Ă— đ?‘?). We will call composed relation

đ?›ź, đ?›˝ đ?‘ƒ ° đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘? đ?œ†, đ?œŒ

đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ)

) to the one defined

by đ?›ź, đ?›˝ đ?‘ƒ ° đ?‘… = {⌊(đ?‘Ľ, đ?‘§), đ?‘‡ đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§), đ??ź đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§), đ??š đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§)âŒŞ /đ?‘Ľ ∈ đ?‘‹, đ?‘§ ∈ đ?‘?} đ?‘ƒ ° đ?‘… đ?‘ƒ ° đ?‘… đ?‘ƒ ° đ?‘… đ?œ†, đ?œŒ đ?œ†,đ?œŒ đ?œ†,đ?œŒ đ?œ†,đ?œŒ đ?›ź Where,đ?‘‡ đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§) = đ?‘Ś {đ?›˝[đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§)]}, đ??ź đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§) = đ?›źđ?‘Ś{đ?›˝[đ??źđ?‘… (đ?‘Ľ, đ?‘Ś), đ??źđ?‘ƒ (đ?‘Ś, đ?‘§)]} , đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

đ??š

đ?›ź,đ?›˝ đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

(đ?‘Ľ, đ?‘§) = đ?‘Śđ?œ†{đ?œŒ[đ??šđ?‘… (đ?‘Ľ, đ?‘Ś), đ??šđ?‘ƒ (đ?‘Ś, đ?‘§)]}

Whenever 0 ≤ �

đ?›ź,đ?›˝ đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

(đ?‘Ľ, đ?‘§) + đ??ź

đ?›ź,đ?›˝ đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

(đ?‘Ľ, đ?‘§) + đ??š

đ?›ź,đ?›˝ đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

(đ?‘Ľ, đ?‘§) ≤ 3 ∀(đ?‘Ľ, đ?‘§) ∈ đ?‘‹ Ă— đ?‘? .

The choice of the t-norms and t-conorms đ?›ź, đ?›˝, đ?œ†, đ?œŒ in the previous definition, is evidently conditioned by the fulfilment of 0 ≤ đ?‘‡ đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§) + đ??ź đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§) + đ??š đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§) ≤ 3 ∀(đ?‘Ľ, đ?‘§) ∈ đ?‘‹ Ă— đ?‘? đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

Theorem 3.6: For each, ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ) , đ?‘ƒ ∈ đ??šđ?‘ đ?‘…(đ?‘Œ Ă— đ?‘?) and đ?›ź, đ?›˝, đ?œ†, đ?œŒ any t-norms or t-conorms đ?›ź, đ?›˝ −1 đ?›ź, đ?›˝ (đ?‘ƒ ° đ?‘…) = đ?‘… −1 ° đ?‘ƒâˆ’1 is fulfilled. đ?œ†, đ?œŒ đ?œ†, đ?œŒ Proof: đ?‘‡

đ?›ź,đ?›˝ −1 (đ?‘ƒ ° đ?‘…) đ?œ†,đ?œŒ

(�, �) = �

đ?›ź,đ?›˝ đ?‘ƒ ° đ?‘… đ?œ†,đ?œŒ

(đ?‘Ľ, đ?‘§) = đ?›źđ?‘Ś {đ?›˝[đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§)]} = đ?›źđ?‘Ś {đ?›˝[đ?‘‡đ?‘…−1 (đ?‘Ś, đ?‘Ľ), đ?‘‡ đ?‘ƒâˆ’1 (đ?‘§, đ?‘Ś)]}

= đ?›źđ?‘Ś {đ?›˝[đ?‘‡đ?‘ƒâˆ’1 (đ?‘§, đ?‘Ś), đ?‘‡ đ?‘…−1 (đ?‘Ś, đ?‘Ľ)]} = đ?‘‡

đ?‘… −1

đ?›ź,đ?›˝ ° đ?‘ƒ −1 đ?œ†,đ?œŒ

(đ?‘§, đ?‘Ľ). Similarly đ??ź

đ?›ź,đ?›˝ −1 ° đ?‘…) đ?œ†,đ?œŒ

(đ?‘§, đ?‘Ľ) = đ??ź

(đ?‘ƒ

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đ?‘… −1

đ?›ź,đ?›˝ ° đ?œ†,đ?œŒ

đ?‘ƒ −1

(�, �),

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𝐹

𝛼,𝛽 −1 (𝑃 ° 𝑅) 𝜆,𝜌

(𝑧, 𝑥) = 𝐹

𝛼,𝛽 𝑃 ° 𝑅 𝜆,𝜌

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(𝑥, 𝑧) = 𝑦𝜆{𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑃 (𝑦, 𝑧)]} = 𝑦𝜆{𝜌[𝐹𝑅−1 (𝑦, 𝑥), 𝐹 𝑃−1 (𝑧, 𝑦)]}

= 𝑦𝜆{𝜌[𝐹𝑃−1 (𝑧, 𝑦), 𝐹 𝑅−1 (𝑦, 𝑥)]} = 𝐹

𝑅 −1

𝛼,𝛽 ° 𝜆,𝜌

𝑃 −1

(𝑧, 𝑥) ∀(𝑧, 𝑥) ∈ 𝑍 × 𝑋.

𝛼, 𝛽 −1 𝛼, 𝛽 Hence(𝑃 ° 𝑅) = 𝑅 −1 ° 𝑃 −1 𝜆, 𝜌 𝜆, 𝜌 Theorem 3.7: In the conditions of definition 3.5 𝛼, 𝛽 𝛼, 𝛽 If 𝑃1 ≤ 𝑃2 , then 𝑃1 ∘ 𝑅 ≤ 𝑃2 ∘ 𝑅, for every R ∈ FNR 𝜆, 𝜌 𝜆, 𝜌 𝛼, 𝛽 𝛼, 𝛽 If 𝑅1 ≤ 𝑅2 , then 𝑃 ∘ 𝑅1 ≤ 𝑃 ∘ 𝑅2 , for every P ∈ FNR 𝜆, 𝜌 𝜆, 𝜌 𝛼, 𝛽 𝛼, 𝛽 If 𝑃1 ≼ 𝑃2 , then 𝑃1 ∘ 𝑅 ≼ 𝑃2 ∘ 𝑅, for every R ∈ FNR 𝜆, 𝜌 𝜆, 𝜌 𝛼, 𝛽 𝛼, 𝛽 If 𝑅1 ≼ 𝑅2 , then 𝑃 ∘ 𝑅1 ≼ 𝑃 ∘ 𝑅2 , for every P ∈ FNR 𝜆, 𝜌 𝜆, 𝜌 𝛼, 𝛽 𝛼, 𝛽 Let R, P be in FNR (X × X), if 𝑃 ≤ 𝑅 , then 𝑃 ° 𝑃 ≤ 𝑅 ° 𝑅 are verified. 𝜆, 𝜌 𝜆, 𝜌 Proof: 𝑃1 ≤ 𝑃2 then 𝑇𝑃1 (𝑦, 𝑧) ≤ 𝑇𝑃2 (𝑦, 𝑧), 𝐼𝑃1 (𝑦, 𝑧) ≤ 𝐼𝑃2 (𝑦, 𝑧) 𝑎𝑛𝑑 𝐹𝑃1 (𝑦, 𝑧) ≥ 𝐹𝑃2 (𝑦, 𝑧) 𝛼 𝛼 𝑇 𝛼,𝛽 (𝑥, 𝑧) = {𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑃1 (𝑦, 𝑧)]} ≤ {𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑃2 (𝑦, 𝑧)]} = 𝑇 𝛼,𝛽 (𝑥, 𝑧). 𝑦

𝑃1 ° 𝑅 𝜆,𝜌

Similarly 𝐼 𝐹

𝛼,𝛽 𝑃1 ° 𝑅 𝜆,𝜌

𝛼,𝛽 𝑃1 ° 𝑅 𝜆,𝜌

(𝑥, 𝑧) ≤ 𝐼

𝑦

𝛼,𝛽 𝑃2 ° 𝑅 𝜆,𝜌

𝑃2 ° 𝑅 𝜆,𝜌

(𝑥, 𝑧)

𝜆

𝜆

𝑦

𝑦

(𝑥, 𝑧) = {𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑃1 (𝑦, 𝑧)]} ≥ {𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑃2 (𝑦, 𝑧)]} = 𝐹

𝛼,𝛽 𝑃2 ° 𝑅 𝜆,𝜌

(𝑥, 𝑧)

𝛼, 𝛽 𝛼, 𝛽 Therefore 𝑃1 ∘ 𝑅 ≤ 𝑃2 ∘ 𝑅 𝜆, 𝜌 𝜆, 𝜌 (ii), (iii), (iv),(v) can be proved in a similar way.

Theorem 3.8: For any 𝛼, 𝛽, 𝜆 𝑎𝑛𝑑 𝜌 t-norms or t-conorms, 𝑅, 𝑃 ∈ 𝐹𝑁𝑅(𝑌 × 𝑍)𝑎𝑛𝑑 𝑄 ∈ 𝐹𝑁𝑅(𝑋 × 𝑌) 𝛼, 𝛽 𝛼, 𝛽 𝛼, 𝛽 (𝑅 ∨ 𝑃) ∘ 𝑄 ≥ 𝑅 ∘ 𝑄 ∨ 𝑃 ∘ 𝑄 ℎ𝑜𝑙𝑑𝑠 𝜆, 𝜌 𝜆, 𝜌 𝜆, 𝜌 Http://www.granthaalayah.com ©International Journal of Research - GRANTHAALAYAH

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Proof: Starting from the points (vi), (vii), (viii) of the theorem 3.4, we get đ?›ź, đ?›˝ đ?›ź, đ?›˝ (đ?‘… ∨ đ?‘ƒ) ∘ đ?‘„ ≼ đ?‘… ∘ đ?‘„ đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?‘… ∨ đ?‘ƒ ≼ đ?‘… { ⇒ ⇒ (đ?‘… ∨ đ?‘ƒ) ∘ đ?‘„ ≼ đ?‘… ∘ đ?‘„ ∨ đ?‘ƒ ∘ đ?‘„ đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?‘… ∨ đ?‘ƒ ≼ đ?‘ƒ đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?œ†, đ?œŒ (đ?‘… ∨ đ?‘ƒ) ∘ đ?‘„ ≼ đ?‘ƒ ∘ đ?‘„ { đ?œ†, đ?œŒ đ?œ†, đ?œŒ The above theorem determines the sign of the inequality for the distributive property of the composition respecting the union. Next theorem will give us a necessary and sufficient condition for the fulfilment of the equality.

Theorem 3.9: Let R, P be the two elements of FNR (đ?‘Œ Ă— Z), Q ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ) , đ?›ź đ?‘Žđ?‘›đ?‘‘ đ?œ† not null t-norms or đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?›ź, đ?›˝ t -conorms. Then (đ?‘… ∨ đ?‘ƒ) ∘ đ?‘„ = đ?‘… ∘ đ?‘„ ∨ đ?‘ƒ ∘ đ?‘„ if and only if đ?›ź = ∨ đ?‘Žđ?‘›đ?‘‘ đ?œ† = ∧. đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?œ†, đ?œŒ Proof: đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?›ź Let(đ?‘… ∨ đ?‘ƒ) ∘ đ?‘„ = đ?‘… ∘ đ?‘„ ∨ đ?‘ƒ ∘ đ?‘„ .đ?‘‡ {đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ś, đ?‘§) ∨ đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§)]} đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘Ľ) = đ?‘Ś (đ?‘…∨đ?‘ƒ) ∘ đ?‘„ đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?œ†,đ?œŒ đ?›ź = đ?‘Ś{đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ś, đ?‘§)] ∨ đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§)]}.Because of the hypothesis of the theorem the result is =

� �

�

{đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ś, đ?‘§)]} ∨ đ?‘Ś{đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§)]}.

Let {đ?‘Žđ?‘Ś }đ?‘Śâˆˆđ?‘Œ be any two finite family of numbers belonging to the interval [0, 1] If đ?›˝ is t-norm, we define (for x, z fixed and for every y). đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś) = 1, đ?‘‡đ?‘… (đ?‘Ś, đ?‘§) = đ?‘Žđ?‘Ś , đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§) = đ?‘?đ?‘Ś .Then it is known that đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ś, đ?‘§)] = đ?‘Žđ?‘Ś đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§)] = đ?‘?đ?‘Ś .Besides as (đ?‘… ∨ đ?‘ƒ) ∘ đ?‘„ = (đ?‘… ∘ đ?‘„) ∨ (đ?‘ƒ ∘ đ?‘„) it is verified đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?œ†, đ?œŒ for every R, P and Q by means of hypothesis we have đ?›źđ?‘–(đ?‘Žđ?‘– ∨ đ?‘?đ?‘– ) = đ?›źđ?‘–(đ?‘Žđ?‘– ) ∨ đ?›źđ?‘–(đ?‘?đ?‘– ) and this condition verified for every {đ?‘Žđ?‘Ś }đ?‘Śâˆˆđ?‘Œ , {đ?‘?đ?‘Ś }đ?‘Śâˆˆđ?‘Œ .We have proved in the theorem 2.0, that is equivalent to đ?›ź = ∨ (ii) If đ?›˝ is t-conorm, we define the degree of truthfulness of R, P and Q as follows: đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś) = 0, đ?‘‡đ?‘… (đ?‘Ś, đ?‘§) = đ?‘Žđ?‘Ś , đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§) = đ?‘?đ?‘Ś and with the same proceeding we conclude that verifying đ?›źđ?‘–(đ?‘Žđ?‘– ∨ đ?‘?đ?‘– ) = đ?›źđ?‘–(đ?‘Žđ?‘– ) ∨ đ?›źđ?‘–(đ?‘?đ?‘– ) as we have seen in the theorem 2.0, if and only if đ?›ź = ∨.

In the same way we can prove the result for indeterminacy and by following the same proceeding for the falseness we can conclude that = ∧ . Conversely,Let đ?›ź = ∨ đ?‘Žđ?‘›đ?‘‘ đ?œ† = ∧, đ?›˝ đ?‘Žđ?‘›đ?‘‘ đ?œŒ be any t∨ norms and t-conorms đ?‘‡ đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§) = {đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ś, đ?‘§) ∨ đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§)]} đ?‘Ś (đ?‘…∨đ?‘ƒ)

∨

∘ đ?‘„ đ?œ†,đ?œŒ

= đ?‘Ś{đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ś, đ?‘§)] ∨ đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§)]}

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Using the associative property of the t-conorms, we ∨ ∨ = đ?‘Ś{đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ś, đ?‘§)]} ∨ đ?‘Ś{đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘§)]} = đ?‘‡ đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§) ∨ đ?‘‡ đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘§) đ?‘… ∘ đ?‘„ đ?œ†,đ?œŒ

∀ (đ?‘Ľ, đ?‘§) ∈ đ?‘‹ Ă— đ?‘? đ??š

(đ?‘…∨đ?‘ƒ)

đ?›ź,đ?›˝ ∘ đ?‘„ đ?œ†,đ?œŒ

đ??ź

.Similarly

(đ?‘Ľ, đ?‘§) = đ??š

đ?›ź,đ?›˝ đ?‘… ∘ đ?‘„ đ?œ†,đ?œŒ

(đ?‘Ľ, đ?‘§) ∧ đ??š

đ?›ź,đ?›˝ đ?‘ƒ ∘ đ?‘„ đ?œ†,đ?œŒ

(đ?‘…∨đ?‘ƒ)

đ?›ź,đ?›˝ ∘ đ?‘„ đ?œ†,đ?œŒ

(đ?‘Ľ, đ?‘§) = đ??ź

have

đ?‘ƒ ∘ đ?‘„ đ?œ†,đ?œŒ

đ?›ź,đ?›˝ đ?‘… ∘ đ?‘„ đ?œ†,đ?œŒ

(đ?‘Ľ, đ?‘§) ∨ đ??ź

đ?›ź,đ?›˝ đ?‘ƒ ∘ đ?‘„ đ?œ†,đ?œŒ

(�, �) and

(�, �)

Theorem 3.10: For every đ?›ź, đ?›˝, đ?œ†, đ?œŒ any t-norms or t-conorms and đ?‘…, đ?‘ƒ ∈ đ??šđ?‘ đ?‘…(đ?‘Œ Ă— đ?‘?) , đ?‘„ ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ), it is đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?›ź, đ?›˝ verified that (đ?‘… ∧ đ?‘ƒ) ∘ đ?‘„ ≤ (đ?‘… ∘ đ?‘„) ∧ (đ?‘ƒ ∘ đ?‘„) đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?œ†, đ?œŒ Proof: Analogous to the one made in the theorem 3.7.

Theorem 3.11: Let R, P be the two elements of FNR (đ?‘Œ Ă— Z), Q ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ) đ?›ź different from the null t-norm đ?‘Žđ?‘›đ?‘‘ đ?œ† different from the null t-conorm. Then đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?›ź, đ?›˝ (đ?‘… ∧ đ?‘ƒ) ∘ đ?‘„ = (đ?‘… ∘ đ?‘„) ∧ (đ?‘ƒ ∘ đ?‘„) if and only if đ?›ź = ∧ đ?‘Žđ?‘›đ?‘‘ đ?œ† = ∨. đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?œ†, đ?œŒ Proof: The proof follows by theorem 3.9.

From the analysis of the previous theorem it is deduced that the choice of đ?›ź, đ?›˝, đ?œ† đ?‘Žđ?‘›đ?‘‘ đ?œŒ t-norms or t-conorms will depend on the problem traced on each case. However the distributive equalities will demand the choice of ∨ đ?‘Žđ?‘›đ?‘‘ ∧ đ?‘“đ?‘œđ?‘&#x; đ?›ź đ?‘Žđ?‘›đ?‘‘ đ?œ† đ?‘œđ?‘&#x; đ?œ† đ?‘Žđ?‘›đ?‘‘ đ?›ź respectively.

Theorem 3.12: Let Q ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ) , P ∈ đ??šđ?‘ đ?‘…(đ?‘Œ Ă— đ?‘?), R ∈ đ??šđ?‘ đ?‘…(đ?‘? Ă— đ?‘?) đ?›˝ đ?‘Žđ?‘›đ?‘‘ đ?œŒ any t-norms or t-conorms. ∨, đ?›˝ ∨, đ?›˝ ∨, đ?›˝ ∨, đ?›˝ If đ?›ź = ∨ đ?‘Žđ?‘›đ?‘‘ đ?œ† = ∧ then (đ?‘… ∘ đ?‘ƒ) ∘ đ?‘„ = đ?‘… ∘ (đ?‘ƒ ∘ đ?‘„) ∧, đ?œŒ ∧, đ?œŒ ∧, đ?œŒ ∧, đ?œŒ Proof: Let đ?›˝ be associative

∨∨ đ?‘– đ?‘— đ?‘Žđ?‘–,đ?‘—

=

∨∨ đ?‘— đ?‘–đ?‘Žđ?‘–,đ?‘—

, đ?›˝(đ?‘Ž, â‹ đ?‘–đ?‘?đ?‘– ) =

and the same properties are applied to ∧.đ?‘‡

∨,đ?›˝ ∨,đ?›˝ (đ?‘… ∘ đ?‘„) ∘ đ?‘„ ∧,đ?œŒ

∧,đ?œŒ

â‹

(đ?›˝(đ?‘Ž, đ?‘?đ?‘– )),đ?›˝( â‹ đ?‘–đ?‘Žđ?‘– , đ?‘? ) = đ?‘–

(�, �) =

∨ đ?‘Ś

{� [�� (�, �), �

â‹ đ?‘–

(đ?›˝(đ?‘Žđ?‘–, đ?‘?))

∨,đ?›˝ đ?‘… ∘ đ?‘ƒ ∧,đ?œŒ

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(�, �) ]}

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[Arockiarani et. al., Vol.4 (Iss.2): February, 2016]

= = =

∨ đ?‘Ś

∨

{đ?›˝ [đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘Ą {đ?›˝(đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘Ą), (đ?‘‡đ?‘… (đ?‘Ą, đ?‘§)) )} ]} =

∨ ∨ đ?‘Ą đ?‘Ś ∨ đ?‘Ą

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{đ?›˝ [đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘Ą), đ?‘‡đ?‘… (đ?‘Ą, đ?‘§)]]} = (đ?‘Ľ, đ?‘Ą), đ?‘‡đ?‘… (đ?‘Ą, đ?‘§)]} = đ?‘‡

{� [�

∨,đ?›˝ đ?‘ƒ ∘ đ?‘„ ∧,đ?œŒ

The equality đ??ź

∨,đ?›˝ ∨,đ?›˝ đ?‘… ∘ (đ?‘ƒ ∘ đ?‘„) ∧,đ?œŒ ∧,đ?œŒ

∨,đ?›˝ ∨,đ?›˝ (đ?‘… ∘ đ?‘ƒ) ∘ đ?‘„ ∧,đ?œŒ

∨,đ?›˝ ∨,đ?›˝ (đ?‘Ľ, đ?‘§) (đ?‘… ∘ đ?‘ƒ) ∘ đ?‘„

đ??š

∧,đ?œŒ

∧,đ?œŒ

(đ?‘Ľ, đ?‘§) = đ??ź

∧,đ?œŒ

= đ??š

∨,đ?›˝ ∨,đ?›˝ đ?‘… ∘ (đ?‘ƒ ∘ đ?‘„) ∧,đ?œŒ ∧,đ?œŒ

∨,đ?›˝ ∨,đ?›˝ đ?‘… ∘ (đ?‘ƒ ∘ đ?‘„) ∧,đ?œŒ ∧,đ?œŒ

∨ đ?‘Ś

∨ ∨ đ?‘Ą {đ?›˝[ đ?‘Ś

{ ∨đ?‘Ą {đ?›˝[đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?›˝(đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘Ą), đ?‘‡đ?‘… (đ?‘Ą, đ?‘§)) ]}} đ?›˝ [đ?‘‡đ?‘„ (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘ƒ (đ?‘Ś, đ?‘Ą), đ?‘‡đ?‘… (đ?‘Ą, đ?‘§) ]]}

(đ?‘Ľ, đ?‘§) ∀ (đ?‘Ľ, đ?‘§) ∈ đ?‘‹ Ă— đ?‘?

(đ?‘Ľ, đ?‘§) đ?‘“đ?‘œđ?‘&#x; đ?‘’đ?‘Łđ?‘’đ?‘&#x;đ?‘Ś (đ?‘Ľ, đ?‘§) ∈ đ?‘‹ Ă— đ?‘? and

(đ?‘Ľ, đ?‘§) ∀ (đ?‘Ľ, đ?‘§) ∈ đ?‘‹ Ă— đ?‘? corresponding to the indeterminacy and

falsity are proved in a similar way.

Note: It is convenient to state that in the notation used for composition

đ?›ź, đ?›˝ đ?‘ƒ ° đ?‘… , đ?œ†, đ?œŒ

the symbols � and �

placed up are applied to the truth value and indeterministic value and the symbols đ?œ† and đ?œŒ placed down are applied to the false value .Therefore the order of placement is very important.

4. RELATIONS ON FUZZY NEUTROSOPHIC SETS Definition 4.1: The relation Δ ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) is called the relation of identity if ∀ (đ?‘Ľ, đ?‘Ś) ∈ đ?‘‹ Ă— đ?‘‹ 1 đ?‘–đ?‘“ đ?‘Ľ = đ?‘Ś 1 đ?‘–đ?‘“ đ?‘Ľ = đ?‘Ś 0 đ?‘–đ?‘“ đ?‘Ľ = đ?‘Ś đ?‘‡âˆ† (đ?‘Ľ, đ?‘Ś) = { , đ??ź (đ?‘Ľ, đ?‘Ś) = { , đ??š (đ?‘Ľ, đ?‘Ś) = { 0 đ?‘–đ?‘“ đ?‘Ľ ≠đ?‘Ś ∆ 0 đ?‘–đ?‘“ đ?‘Ľ ≠đ?‘Ś ∆ 1 đ?‘–đ?‘“ đ?‘Ľ ≠đ?‘Ś The complementary relation ∆đ?‘? = ∇ is defined by 0 đ?‘–đ?‘“ đ?‘Ľ = đ?‘Ś 0 đ?‘–đ?‘“ đ?‘Ľ = đ?‘Ś 1 đ?‘–đ?‘“ đ?‘Ľ = đ?‘Ś , đ??źâˆ‡ (đ?‘Ľ, đ?‘Ś) = { , đ??šâˆ‡ (đ?‘Ľ, đ?‘Ś) = { .

1 đ?‘–đ?‘“ đ?‘Ľ ≠đ?‘Ś 1 đ?‘–đ?‘“ đ?‘Ľ ≠đ?‘Ś 0 đ?‘–đ?‘“ đ?‘Ľ ≠đ?‘Ś It is evident that ∆ = ∆−1 and ∇ = ∇−1 đ?‘‡âˆ‡ (đ?‘Ľ, đ?‘Ś) = {

Theorem 4.2: Let đ?›ź, đ?›˝, đ?œ†, đ?œŒ be t-norms and t-conorms and đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?‘… ∘ ∆ = ∆ ∘ đ?‘… = đ?‘… if and only if đ?›ź is t-conorm , đ?›˝ is t-norm , đ?œ† is t-norm and đ?œŒ is tđ?œ†, đ?œŒ đ?œ†, đ?œŒ conorm. đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?‘… ∘ ∇ = ∇ ∘ đ?‘… = đ?‘… if and only if đ?›ź is t-conorm , đ?›˝ is t-norm , đ?œ† is t-norm and đ?œŒ is tđ?›ź, đ?›˝ đ?›ź, đ?›˝ conorm.

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Proof: 𝛼 𝛼 𝛼,𝛽 (𝑥, 𝑧) = 𝑦{𝛽[𝑇∆ (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)]} = 𝑦≠𝑥{𝛽[𝑇∆ (𝑥, 𝑥), 𝑇𝑅 (𝑥, 𝑧)], 𝛽[𝑇∆ (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)] 𝑅 ∘ ∆ 𝜆,𝜌 = 𝑦≠𝑥𝛼{𝛽[1, 𝑇𝑅 (𝑥, 𝑧)], 𝛽[0, 𝑇𝑅 (𝑦, 𝑧)] } = 𝑦≠𝑥𝛼{[𝑇𝑅 (𝑥, 𝑧), 0] } = 𝑇𝑅 (𝑥, 𝑧) , (𝑥, 𝑧) ∈ 𝑋 × 𝑋

𝑇

}

𝛼,𝛽 (𝑥, 𝑧) = 𝐼𝑅 (𝑥, 𝑧) , (𝑥, 𝑧) ∈ 𝑋 × 𝑋 𝑅 ∘ ∆ 𝜆,𝜌 𝐹 𝛼,𝛽 (𝑥, 𝑧) = 𝑦𝜆{𝜌[𝐹∆ (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)]} = 𝑦≠𝑥𝜆{𝜌[𝐹∆ (𝑥, 𝑥), 𝐹𝑅 (𝑥, 𝑧)], 𝜌[𝐹∆ (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)] 𝑅 ∘ ∆ 𝜆,𝜌 = 𝑦≠𝑥𝜆{𝜌[0, 𝐹𝑅 (𝑥, 𝑧)], 𝜌[1, 𝐹𝑅 (𝑦, 𝑧)] } = 𝑦≠𝑥𝜆{[𝐹𝑅 (𝑥, 𝑧), 1] } = 𝐹𝑅 (𝑥, 𝑧) , (𝑥, 𝑧) ∈ 𝑋 × 𝑋

}

Similarly, 𝐼

𝛼, 𝛽 𝛼, 𝛽 Conversely, Suppose that 𝑅 ∘ ∆ = ∆ ∘ 𝑅 = 𝑅 it is fulfilled for each 𝑅 ∈ 𝐹𝑁𝑅(𝑋 × 𝑋) 𝜆, 𝜌 𝜆, 𝜌 Case (i): 𝛼, 𝛽 Let 𝛼 be t-norm and 𝛽 be t-conorm .Taking 𝑅 = Δ , we get Δ ∘ ∆ = ∆ then 𝑇 𝛼,𝛽 (𝑥, 𝑧) = ∆ ∘ ∆ 𝜆, 𝜌 𝜆,𝜌

𝑇∆ (𝑥, 𝑧) , ∀(𝑥, 𝑧) ∈ 𝑋 × 𝑋.If 𝑇 𝛼,𝛽 (𝑥, 𝑥) = 𝛼𝑦{𝛽[𝑇∆ (𝑥, 𝑦), 𝑇∆ (𝑦, 𝑥)]} = ∆ ∘ ∆ 𝜆,𝜌 = 𝑦≠𝑥𝛼{𝛽[1,1], 𝛽[0,0]

𝛼 𝑦≠𝑥{[1,0]

}=

𝑥=𝑧

then

𝛼 𝑦≠𝑥{𝛽[𝑇∆ (𝑥, 𝑥), 𝑇∆ (𝑥, 𝑥)], 𝛽[𝑇∆ (𝑥, 𝑦), 𝑇∆ (𝑦, 𝑥)]

}

} = 0 ≠ 𝑇∆ (𝑥, 𝑥) = 1 ∀𝑥 ∈ 𝑋 Which is a contradiction.

Case (ii): Let 𝛼 be t-conorm and 𝛽 be t-conorm .Taking 𝑅 = Δ and 𝑥 ≠ 𝑧 we have 𝑇 𝛼,𝛽 (𝑥, 𝑧) = 𝛼𝑦{𝛽[𝑇∆ (𝑥, 𝑦), 𝑇∆ (𝑦, 𝑧)]} = 𝑦≠𝑥𝛼{𝛽[𝑇∆ (𝑥, 𝑥), 𝑇∆ (𝑥, 𝑧)], 𝛽[𝑇∆ (𝑥, 𝑦), 𝑇∆ (𝑦, 𝑧)] } ∆ ∘ ∆ 𝜆,𝜌 = 𝑦≠𝑥𝛼{𝛽[1, 𝑇∆ (𝑥, 𝑧)], 𝛽[0, 𝑇∆ (𝑦, 𝑧)]

=

𝛼 𝑦≠𝑥{𝛽[1,0], 𝛽[0,1], 𝛽[0,0] 𝑦≠𝑧

For every (𝑥, 𝑧) ∈ 𝑋 × 𝑋, 𝑇

}=

}=

𝛼 𝑦≠𝑥{𝛽[1,0], 𝛽[0, 𝑇∆ (𝑧, 𝑧)], 𝛽[0, 𝑇∆ (𝑦, 𝑧)] 𝑦≠𝑧

𝛼 𝑦≠𝑥{1,1,0 𝑦≠𝑧

𝛼,𝛽 (𝑥, 𝑧) ∆ ∘ ∆ 𝜆,𝜌

}

} = 1 ≠ 𝑇∆ (𝑥, 𝑧) = 0 ∀ (𝑥, 𝑧) ∈ 𝑋 × 𝑋

≠ 𝑇∆ (𝑥, 𝑧) if 𝑥 ≠ 𝑧

Case (iii): 1 𝑖𝑓 𝑥 = 𝑦 . ≠ 1 𝑖𝑓 𝑥 ≠ 𝑦 (𝑥, 𝑥) = 𝑇𝑅 (𝑥, 𝑥) = 1 ∀ 𝑥 ∈ 𝑋 have to be fulfilled.

Let 𝛼 be t-norm and 𝛽 be t-conorm. Taking 𝑅 in the following way 𝑇𝑅 (𝑥, 𝑦) = { By means of hypothesis 𝑇

𝛼,𝛽 𝑅 ∘ ∆ 𝜆,𝜌

𝛼 𝛼 𝛼,𝛽 (𝑥, 𝑥) = 𝑦 {𝛽[𝑇∆ (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑥)]} = 𝑦≠𝑥{𝛽[𝑇∆ (𝑥, 𝑥), 𝑇𝑅 (𝑥, 𝑥)], 𝛽[𝑇∆ (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑥)] } 𝑅 ∘ ∆ 𝜆,𝜌 = 𝑦≠𝑥𝛼{𝛽[1,1], 𝛽[0, 𝑇𝑅 (𝑦, 𝑥)] } = 𝑦≠𝑥𝛼{1, 𝑇𝑅 (𝑦, 𝑥) } = 𝑇𝑅 (𝑦, 𝑥) ≠ 1 = 𝑇𝑅 (𝑥, 𝑥) which is not true.

𝑇

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Case (iv): 1 đ?‘–đ?‘“ đ?‘Ľ = đ?‘Ś ≠1 đ?‘–đ?‘“ đ?‘Ľ ≠đ?‘Ś (đ?‘Ľ, đ?‘Ľ) = đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ľ) = 1 ∀ đ?‘Ľ ∈ đ?‘‹ have to be fulfilled.

Let � be t-conorm and � be t-norm .Taking � in the following way �� (�, �) = { by means of hypothesis �

đ?›ź,đ?›˝ đ?‘… ∘ ∆ đ?œ†,đ?œŒ

đ?›ź đ?›ź đ?›ź,đ?›˝ (đ?‘Ľ, đ?‘Ľ) = đ?‘Ś {đ?›˝[đ?‘‡âˆ† (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ś, đ?‘Ľ)]} = đ?‘Śâ‰ đ?‘Ľ{đ?›˝[đ?‘‡âˆ† (đ?‘Ľ, đ?‘Ľ), đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ľ)], đ?›˝[đ?‘‡âˆ† (đ?‘Ľ, đ?‘Ś), đ?‘‡đ?‘… (đ?‘Ś, đ?‘Ľ)] đ?‘… ∘ ∆ đ?œ†,đ?œŒ = đ?‘Śâ‰ đ?‘Ľđ?›ź{đ?›˝[1,1], đ?›˝[0, đ?‘‡đ?‘… (đ?‘Ś, đ?‘Ľ)] } = đ?‘Śâ‰ đ?‘Ľđ?›ź{1,0 } = 1 = đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ľ).

�

}

Hence it is proved that � is t-conorm and � is t-norm. Proof for indeterministic functions is similar to the above.

Making a development, which is analogous to the previous one, for the falsity functions ,we deduce that đ?œ† is t-norm and đ?œŒ is t-conorm.

The proof of (ii) is similar to the one made in (i) using ∇ .

Definition 4.3: The relation đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ) is called Reflexive if for every đ?‘Ľ ∈ đ?‘‹ , đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ľ) = 1 , đ??źđ?‘… (đ?‘Ľ, đ?‘Ľ) = 1, đ??šđ?‘… (đ?‘Ľ, đ?‘Ľ) = 0 Anti- reflexive if for every đ?‘Ľ ∈ đ?‘‹ , đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ľ) = 0 , đ??źđ?‘… (đ?‘Ľ, đ?‘Ľ) = 0, đ??šđ?‘… (đ?‘Ľ, đ?‘Ľ) = 1 (i.e,) The relation đ?‘… is called anti-reflexive if its complement đ?‘… đ?‘? is reflexive.

Theorem 4.4: For every đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ), it is verified that If đ?‘… is reflexive then ∆ ≤ đ?‘… (ii) If đ?‘… is anti-reflexive then ∆ ≼ đ?‘… Proof: It is the consequence of the definition 4.1 and 4.3.

Theorem 4.5: đ?›ź t-conorm, đ?›˝ t-norm, đ?œ† t-norm and đ?œŒ t-conorm it is verified that đ?›ź, đ?›˝ If đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ) is reflexive, then đ?‘… ≤ đ?‘… ∘ đ?‘… đ?œ†, đ?œŒ đ?œ†, đ?œŒ If đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘Œ) is anti-reflexive, then đ?‘… ≼ đ?‘… ∘ đ?‘… đ?›ź, đ?›˝ For

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Proof: 𝛼 𝛼,𝛽 (𝑥, 𝑧) = 𝑦 {𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)]} 𝑅 ∘ 𝑅 𝜆,𝜌 = 𝑦≠𝑥𝛼{𝛽[1, 𝑇𝑅 (𝑥, 𝑧)], 𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)] }

𝑇

because 𝛼 is t-conorm .Similarly 𝐼

𝛼,𝛽 𝑅 ∘ 𝑅 𝜆,𝜌

𝛼 𝑦≠𝑥{𝛽[𝑇𝑅 (𝑥, 𝑥), 𝑇𝑅 (𝑥, 𝑧)], 𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)]

}

= 𝑦≠𝑥𝛼{𝑇𝑅 (𝑥, 𝑧), 𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)] } ≥ 𝑇𝑅 (𝑥, 𝑧) (𝑥, 𝑧) ≥ 𝐼𝑅 (𝑥, 𝑧)

𝜆 𝛼,𝛽 (𝑥, 𝑧) = 𝑦{𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)]} 𝑅 ∘ 𝑅 𝜆,𝜌 = 𝑦≠𝑥𝜆{𝜌[0, 𝐹𝑅 (𝑥, 𝑧)], 𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)] }

𝐹

=

=

𝜆 𝑦≠𝑥{𝜌[𝐹𝑅 (𝑥, 𝑥), 𝐹𝑅 (𝑥, 𝑧)], 𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)]

=

𝜆 𝑦≠𝑥{𝐹𝑅 (𝑥, 𝑧), 𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)]

}

} ≤ 𝐹𝑅 (𝑥, 𝑧)

since 𝜆 is t-norm.

The proof of (ii) is analogous to the proof of (i) .

Example 4.6: This example states the existence of fuzzy neutrosophic relations which satisfy the property 𝛼, 𝛽 𝑅 ≤ 𝑅 ∘ 𝑅 and they are not reflexive.Let 𝑋 be the following set 𝑋 = {𝑥, 𝑦, 𝑧} and 𝜆, 𝜌 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 𝑥 0.4 0.8 0.4 𝑥 0.3 0.7 0.2 𝑅 ∈ 𝐹𝑁𝑅(𝑋 × 𝑌) given by 𝑇𝑅 = (𝑦 ) , 𝐼𝑅 = (𝑦 ), 0.7 0.9 0.6 0.5 0.8 0.5 𝑧 0.3 0.5 0.2 𝑧 0.1 0.4 0.1 𝑥 𝑦 𝑧 𝑥 0.5 0.3 0.9 𝐹𝑅 = (𝑦 ) . For 𝛼 =∨, 𝛽 =∧, 𝜆 =∧ and 𝜌 =∨, we have 0.4 0 0.5 𝑧 0.8 0.4 0.6 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 𝑥 0.7 0.8 0.6 𝑥 0.5 0.7 0.5 𝑇 ∨,∧ = (𝑦 ) , 𝐼 ∨,∧ = (𝑦 ), 0.7 0.9 0.6 0.5 0.8 0.5 𝑅 ∘ 𝑅 𝑅 ∘ 𝑅 ∧,∨ ∧,∨ 𝑧 0.5 0.5 0.5 𝑧 0.4 0.4 0.4 𝑥 𝑦 𝑧 ∨,∧ 𝑥 0.4 0.3 0.5 𝐹 ∨,∧ = (𝑦 ) Resulting that 𝑅 ≤ 𝑅 ∘ 𝑅 not being 𝑅 reflexive.

0.4 0 0.5 𝑅 ∘ 𝑅 ∧,∨ ∧,∨ 𝑧 0.4 0.4 0.5 Theorem 4.7: If 𝑅 ∈ 𝐹𝑁𝑅(𝑋 × 𝑋) is reflexive, 𝛼 , 𝛽 are t-conorms and 𝜆, 𝜌 are t-norms ,then 𝛼, 𝛽 𝑅 ≤ 𝑅 ∘ 𝑅. 𝜆, 𝜌

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Proof: 𝑇 𝛼,𝛽 (𝑥, 𝑧) = 𝛼𝑦 {𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)]} = 𝑅 ∘ 𝑅 𝜆,𝜌 𝛼 {𝛽[1, 𝑇𝑅 (𝑥, 𝑧)], 𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)] 𝑦≠𝑥

= Similarly we can prove 𝐼

𝛼,𝛽 𝑅 ∘ 𝑅 𝜆,𝜌

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𝛼 𝑦≠𝑥{𝛽[𝑇𝑅 (𝑥, 𝑥), 𝑇𝑅 (𝑥, 𝑧)], 𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)]

}

} = 𝑦≠𝑥𝛼{1, 𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)] } = 1 ≥ 𝑇𝑅 (𝑥, 𝑧) (𝑥, 𝑧) ≥ 𝐼𝑅 (𝑥, 𝑧).

𝜆 𝛼,𝛽 (𝑥, 𝑧) = 𝑦{𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)]} 𝑅 ∘ 𝑅 𝜆,𝜌 𝜆{𝜌[0, = 𝑦≠𝑥 𝐹𝑅 (𝑥, 𝑧)], 𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)] }

𝐹

=

𝜆 𝑦≠𝑥{𝜌[𝐹𝑅 (𝑥, 𝑥), 𝐹𝑅 (𝑥, 𝑧)], 𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)]

=

𝜆 𝑦≠𝑥{0, 𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑧)]

}

} = 0 ≤ 𝐹𝑅 (𝑥, 𝑧)

𝛼, 𝛽 Therefore 𝑅 ≤ 𝑅 ∘ 𝑅. 𝜆, 𝜌 Theorem 4.8: Given 𝑅 ∈ 𝐹𝑁𝑅(𝑋 × 𝑌) for 𝛼 t-conorm and 𝜆 t-norm, it is verified that (i) If 𝑅 is reflexive 𝛼, 𝛽 𝛼, 𝛽 then 𝑅 ∘ 𝑅 is reflexive (ii) If 𝑅 is anti-reflexive then 𝑅 ∘ 𝑅 is anti reflexive 𝜆, 𝜌 𝜆, 𝜌 Proof: 𝛼 𝛼 𝛼,𝛽 (𝑥, 𝑥) = 𝑦 {𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑥)]} = 𝑦≠𝑥{𝛽[𝑇𝑅 (𝑥, 𝑥), 𝑇𝑅 (𝑥, 𝑥)], 𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑥)] ∘ 𝑅 𝑅 𝜆,𝜌 𝛼 𝛼 𝑦≠𝑥{𝛽[1,1], 𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑥)] } = 𝑦≠𝑥{1, 𝛽[𝑇𝑅 (𝑥, 𝑦), 𝑇𝑅 (𝑦, 𝑧)] } = 1

𝑇

= Similarly we can prove 𝐼

𝛼,𝛽 (𝑥, 𝑧) 𝑅 ∘ 𝑅 𝜆,𝜌

}

=1

𝜆 𝜆 𝛼,𝛽 (𝑥, 𝑥) = 𝑦{𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑥)]} = 𝑦≠𝑥{𝜌[𝐹𝑅 (𝑥, 𝑥), 𝐹𝑅 (𝑥, 𝑥)], 𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑥)] ∘ 𝑅 𝑅 𝜆,𝜌 = 𝑦≠𝑥𝜆{𝜌[0,0], 𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑥)] } = 𝑦≠𝑥𝜆{0, 𝜌[𝐹𝑅 (𝑥, 𝑦), 𝐹𝑅 (𝑦, 𝑥)] } = 0.Therefore

𝐹

}

𝛼, 𝛽 𝑅 ∘ 𝑅 is reflexive. 𝜆, 𝜌 Proof of (ii) is similar to the one made for the reflexivity.

Corollary 4.9: 𝑅 ∈ 𝐹𝑁𝑅(𝑋 × 𝑋)

If

is

reflexive

𝛼

is

t-norm

and

𝜆

is

t-conorm

,then

𝑛 𝑡𝑖𝑚𝑒𝑠

𝑅

(𝑛)

⏞ 𝛼, 𝛽 𝛼, 𝛽 𝛼, 𝛽 = 𝑅 ∘ 𝑅 ∘ 𝑅 … … . ∘ 𝑅 with n=1,2,3,...... it is reflexive. 𝜆, 𝜌 𝜆, 𝜌 𝜆, 𝜌

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Theorem 4.10: Let đ?‘…1 be reflexive fuzzy neutrosophic relation in đ?‘‹ Ă— đ?‘‹ .Then (i) (đ?‘…1 )−1 is reflexive. đ?‘…1 ∨ đ?‘…2 is reflexive for every đ?‘…2 ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) (iii) đ?‘…1 ∧ đ?‘…2 is reflexive ⇔ đ?‘…2 ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) is reflexive.

(ii)

Proof: Proof follows from the definitions.

Definition 4.11: A reflexive closure of a relation is reflexive for every đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) is defined as đ?‘… ∨ ∆ Definition 4.12: A relation đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) is called symmetric if đ?‘… = đ?‘… −1 that is, if for every (đ?‘Ľ, đ?‘Ś) of đ?‘‹ Ă— đ?‘‹, đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ś) = đ?‘‡đ?‘… (đ?‘Ś, đ?‘Ľ), đ??źđ?‘… (đ?‘Ľ, đ?‘Ś) = đ??źđ?‘… (đ?‘Ś, đ?‘Ľ), đ??šđ?‘… (đ?‘Ľ, đ?‘Ś) = đ??šđ?‘… (đ?‘Ś, đ?‘Ľ).

In a contrary manner we will say that it is asymmetric.

Definition 4.13: Let đ?‘… be an element of đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹).We will say that it is antisymmetrical fuzzy neutrosophic relation if for every (đ?‘Ľ, đ?‘Ś) of đ?‘‹ Ă— đ?‘‹, đ?‘Ľ ≠đ?‘Ś ⇒ đ?‘‡đ?‘… (đ?‘Ľ, đ?‘Ś) ≠đ?‘‡đ?‘… (đ?‘Ś, đ?‘Ľ), đ??źđ?‘… (đ?‘Ľ, đ?‘Ś) ≠đ??źđ?‘… (đ?‘Ś, đ?‘Ľ), đ??šđ?‘… (đ?‘Ľ, đ?‘Ś) ≠đ??šđ?‘… (đ?‘Ś, đ?‘Ľ).

Theorem 4.14: If đ?›ź, đ?›˝, đ?œ†, đ?œŒ are either t-norms or t-conorms and đ?‘…, đ?‘ƒ ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) are symmetrical then đ?›ź, đ?›˝ đ?›ź, đ?›˝ −1 đ?‘… ∘ đ?‘ƒ = (đ?‘ƒ ∘ đ?‘…) đ?œ†, đ?œŒ đ?œ†, đ?œŒ Proof: đ?›ź, đ?›˝ đ?›ź, đ?›˝ đ?›ź, đ?›˝ −1 đ?‘… and đ?‘ƒ are symmetrical ⇒ đ?‘… = đ?‘… −1 , đ?‘ƒ = đ?‘ƒâˆ’1 then đ?‘… ∘ đ?‘ƒ = đ?‘… −1 ∘ đ?‘ƒ −1 = (đ?‘ƒ ∘ đ?‘…) . đ?œ†, đ?œŒ đ?œ†, đ?œŒ đ?œ†, đ?œŒ Definition 4.15: Let us take đ?›ź t-conorm, đ?›˝ t-norm, đ?œ† t-norm and đ?œŒ t-conorm, we will say that đ?›ź, đ?›˝ đ?œ†, đ?œŒ đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) is (i) transitive if đ?‘… ≼ đ?‘… ∘ đ?‘… (ii) c-transitive if đ?‘… ≤ đ?‘… ∘ đ?‘…. đ?œ†, đ?œŒ đ?›ź, đ?›˝ Result 4.16: For đ?›ź t-conorm, đ?›˝ t-norm, đ?œ† t-norm and đ?œŒ t-conorm, it is verified that

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(i) If

đ?›ź, đ?›˝ đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) is reflexive and transitive, then đ?‘… = đ?‘… ∘ đ?‘… đ?œ†, đ?œŒ

(ii) If

đ?œ†, đ?œŒ đ?‘… ∈ đ??šđ?‘ đ?‘…(đ?‘‹ Ă— đ?‘‹) is anti-reflexive and c- transitive, then đ?‘… = đ?‘… ∘ đ?‘… đ?›ź, đ?›˝

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