Fuzzy Neutrosophic Relations

2016·0.81 MB·

by  I. Arockiarani

[Arockiarani et. al., Vol.4 (Iss.2): February, 2016] ISSN- 2350-0530(O) ISSN- 2394-3629(P)
Impact Factor: 2.035 (I2OR)

Science
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 t-
conorms 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 t-
conorms.

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 T :
[0,1]  [0, 1]  [0, 1] satisfying the following properties

1. Boundary conditions, T(x, 1) = x and T(x, 0) = 0, for all x  [0, 1]
2. Monotony, T(x, y) ≤ T (z, t) if x ≤ z and y ≤ t
3. Commutative, T(x, y) = T(y, x), for all x, y  [0, 1]
4. Associative, T(T(x, y), z) = T(x, T(y, z)), for all x, y, z  [0, 1]

Given a t-norm T, we can consider the mapping S: [0,1]  [0, 1]  [0, 1] S(x, y) 1-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 𝛼,𝛽,𝜆,𝜌. Let I be a finite
family of indices and {𝑎 } , {𝑏 } be number collection of [0, 1].For every 𝛼 t-norm or t-
𝑖 𝑖∈𝐼 𝑖 𝑖∈𝐼
conorm and for every 𝜆 t-norm or t-conorm

1. 𝛼(𝑎 ∨ 𝑏 ) ≥ 𝛼(𝑎 )∨ 𝛼(𝑏 )
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
2. 𝜆(𝑎 ∧ 𝑏 ) ≤ 𝜆(𝑎 )∧ 𝜆(𝑏 ) are verified.
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖

With the result given by L.W.Fung and S.K.Ku [10] relative to the fact that 𝛼 is an idempotent t-
conorm (idempotent t-norm) if and only if 𝛼 =∨(𝛼 =∧), we get the following theorem:

Theorem 2.0:

Let {𝑎 } , {𝑏 } be two finite number families of [0,1] and 𝛼,𝜆 t-norms or t-conorms not null.
𝑖 𝑖∈𝐼 𝑖 𝑖∈𝐼
Then
𝛼(𝑎 ∨ 𝑏 ) = 𝛼(𝑎 )∨ 𝛼(𝑏 ) if and only if 𝛼 =∨
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
𝜆(𝑎 ∧ 𝑏 ) = 𝜆(𝑎 )∧ 𝜆(𝑏 ) if and only if 𝜆 =∧ .
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖

Definition 2.1: [2]

A Fuzzy neutrosophic set A on the universe of discourse X is defined as
A= 〈𝑥,𝑇 (𝑥),𝐼 (𝑥),𝐹 (𝑥)〉,𝑥 ∈ 𝑋 where 𝑇,𝐼,𝐹:𝑋 →[0, 1] and 0T (x)I (x) F (x)3
𝐴 𝐴 𝐴 A A A

3. FUZZY NEUTROSOPHIC RELATIONS

Definition 3.1:

A fuzzy neutrosophic set relation is defined as a fuzzy neutrosophic subset of 𝑋×𝑌 having the
form 𝑅 = { 〈(𝑥,𝑦),𝑇 (𝑥,𝑦),𝐼 (𝑥,𝑦),𝐹 (𝑥,𝑦)〉:𝑥 ∈ 𝑋,𝑦 ∈ 𝑌 } where 𝑇 ,𝐼 ,𝐹 :𝑋×𝑌 → [0,1]
𝑅 𝑅 𝑅 𝑅 𝑅 𝑅
Satisfy the condition 0 ≤ 𝑇 (𝑥,𝑦)+𝐼 (𝑥,𝑦)+𝐹 (𝑥,𝑦) ≤ 3 ∀ (𝑥,𝑦) ∈ 𝑋 ×𝑌.
𝑅 𝑅 𝑅
We will denote with 𝐹𝑁𝑅(𝑋×𝑌) the set of all fuzzy neutrosophic subsets in 𝑋×𝑌.

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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 𝑇 (𝑦,𝑥) = 𝑇 (𝑥,𝑦),𝐼 (𝑦,𝑥) = 𝐼 (𝑥,𝑦),𝐹 (𝑦,𝑥) = 𝐹 (𝑥,𝑦) ∀(𝑥,𝑦) ∈
𝑅−1 𝑅 𝑅−1 𝑅 𝑅−1 𝑅
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 ⇔ 𝑇 (𝑥,𝑦) ≤ 𝑇 (𝑥,𝑦),𝐼 (𝑥,𝑦) ≤ 𝐼 (𝑥,𝑦),𝐹 (𝑥,𝑦) ≥ 𝐹 (𝑥,𝑦)
𝑅 𝑃 𝑅 𝑃 𝑅 𝑃
2)𝑅 ≼ 𝑃 ⇔ 𝑇 (𝑥,𝑦) ≤ 𝑇 (𝑥,𝑦),𝐼 (𝑥,𝑦) ≤ 𝐼 (𝑥,𝑦),𝐹 (𝑥,𝑦) ≤ 𝐹 (𝑥,𝑦)
𝑅 𝑃 𝑅 𝑃 𝑅 𝑃
3)𝑅 ∨𝑃 = { 〈(𝑥,𝑦),𝑇 (𝑥,𝑦)∨𝑇 (𝑥,𝑦),𝐼 (𝑥,𝑦)∨ 𝐼 (𝑥,𝑦),𝐹 (𝑥,𝑦)∧ 𝐹 (𝑥,𝑦)〉
𝑅 𝑃 𝑅 𝑃 𝑅 𝑃
4)𝑅 ∧𝑃 = { 〈(𝑥,𝑦),𝑇 (𝑥,𝑦)∧𝑇 (𝑥,𝑦),𝐼 (𝑥,𝑦)∧ 𝐼 (𝑥,𝑦),𝐹 (𝑥,𝑦)∨ 𝐹 (𝑥,𝑦)〉
𝑅 𝑃 𝑅 𝑃 𝑅 𝑃
5)𝑅𝑐 = { 〈(𝑥,𝑦),𝐹 (𝑥,𝑦),1−𝐼 (𝑥,𝑦),𝑇 (𝑥,𝑦)〉:𝑥 ∈ 𝑋,𝑦 ∈ 𝑌}
𝑅 𝑅 𝑅

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 𝑇 (𝑦,𝑥) = 𝑇 (𝑥,𝑦) ≤ 𝑇 (𝑥,𝑦) = 𝑇 (𝑦,𝑥) 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦(𝑥,𝑦) ∈ 𝑋×𝑌
𝑅−1 𝑅 𝑃 𝑃−1
𝐼 (𝑦,𝑥) = 𝐼 (𝑥,𝑦) ≤ 𝐼 (𝑥,𝑦) = 𝐼 (𝑦,𝑥) ∀(𝑥,𝑦) ∈ 𝑋 ×𝑌 and
𝑅−1 𝑅 𝑃 𝑃−1
𝐹 (𝑦,𝑥) = 𝐹 (𝑥,𝑦) ≥ 𝐹 (𝑥,𝑦) = 𝐹 (𝑦,𝑥) ∀(𝑥,𝑦) ∈ 𝑋×𝑌 .Hence 𝑅−1 ≤ 𝑃−1
𝑅−1 𝑅 𝑃 𝑃−1
(𝒊𝒊) 𝑇 (𝑦,𝑥) = 𝑇 (𝑥,𝑦) = 𝑇 (𝑥,𝑦)∨ 𝑇 (𝑥,𝑦) = 𝑇 (𝑦,𝑥)∨ 𝑇 (𝑦,𝑥)
(𝑅∨𝑃)−1 𝑅∨𝑃 𝑅 𝑃 𝑅−1 𝑃−1
= 𝑇 (𝑦,𝑥).The proofs for 𝐼 (𝑦,𝑥) = 𝐼 (𝑦,𝑥) &
𝑅−1∨𝑃−1 (𝑅∨𝑃)−1 𝑅−1∨𝑃−1
𝐹 (𝑦,𝑥) = 𝐹 (𝑦,𝑥) are done in a similar way.Hence (𝑅∨𝑃)−1 = 𝑅−1 ∨𝑃−1.
(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].𝑇 (𝑥,𝑦) = 𝑇 (𝑥,𝑦) ∧ {𝑇 (𝑥,𝑦)∨ 𝑇 (𝑥,𝑦)}
𝑅∧(𝑃∨𝑄) 𝑅 𝑃 𝑄
= { 𝑇 (𝑥,𝑦) ∧ 𝑇 (𝑥,𝑦)}∨{ (𝑇 (𝑥,𝑦)∧ 𝑇 (𝑥,𝑦)} = 𝑇 (𝑥,𝑦) ∧ 𝑇 (𝑥,𝑦)
𝑅 𝑃 𝑅 𝑄 𝑅∧𝑃 𝑅∧𝑄
= 𝑇 (𝑥,𝑦) . Similarly 𝐼 (𝑥,𝑦) = 𝐼 (𝑥,𝑦) 𝑎nd 𝐹 (𝑥,𝑦) =
(𝑅∧𝑃) ∨ (𝑅∧𝑄) 𝑅∧(𝑃∨𝑄) (𝑅∧𝑃) ∨ (𝑅∧𝑄) 𝑅∧(𝑃∨𝑄)
𝐹 (𝑥,𝑦). Hence 𝑅 ∧(𝑃∨𝑄) = (𝑅∧𝑃)∨(𝑅∧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 ,𝑇 ≥ 𝑇 𝑎𝑛𝑑 𝑇 ≥ 𝑇 𝐼 ≥ 𝐼 𝑎𝑛𝑑 𝐼 ≥ 𝐼 𝐹 ≤ 𝐹 𝑎𝑛𝑑 𝐹 ≤ 𝐹
𝑅 𝑃 𝑅 𝑄, 𝑅 𝑃 𝑅 𝑄, 𝑅 𝑃 𝑅 𝑄,
⇒ 𝑇 ≥ 𝑇 ∨𝑇 𝐼 ≥ 𝐼 ∨𝐼 𝐹 ≤ 𝐹 ∧𝐹 ⇒ 𝑇 ≥ 𝑇 ,𝐼 ≥ 𝐼 ,𝐹 ≤ 𝐹 ⇒ 𝑅 ≥ 𝑃∨𝑄
𝑅 𝑃 𝑄, 𝑅 𝑃 𝑄 , 𝑅 𝑃 𝑄, 𝑅 𝑃∨𝑄 𝑅 𝑃∨𝑄 𝑅 𝑃∨𝑄
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Similarly we can prove (𝒗𝒊𝒊𝒊) .We can generalize the operations between binary fuzzy
neutrosophic relations R,Q ∈ FNR (X×Y). Using the well known triangular t-norms and t-
conorms in [0, 1]. For a triangular t – norms T and its dual t – conforms S, we get
𝑇(𝑅,𝑄) = { 〈(𝑥,𝑦),𝑇(𝑇 (𝑥,𝑦),𝑇 (𝑥,𝑦),𝑇(𝐼 (𝑥,𝑦),𝐼 (𝑥,𝑦)),𝑆(𝐹 (𝑥,𝑦),𝐹 (𝑥,𝑦))〉 }
𝑅 𝑄 𝑅 𝑄 𝑅 𝑄
𝑆(𝑅,𝑄) = { 〈(𝑥,𝑦),𝑆(𝑇 (𝑥,𝑦),𝑇 (𝑥,𝑦),𝑆(𝐼 (𝑥,𝑦),𝐼 (𝑥,𝑦)),𝑇(𝐹 (𝑥,𝑦),𝐹 (𝑥,𝑦))〉 }
𝑅 𝑄 𝑅 𝑄 𝑅 𝑄

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 𝛼,𝛽,𝜆,𝜌 be t-norms or t-conorms not necessarily dual two – two, 𝑅 ∈ 𝐹𝑁𝑅(𝑋×𝑌)
𝛼,𝛽
and 𝑃 ∈ 𝐹𝑁𝑅(𝑌×𝑍). We will call composed relation 𝑃 ° 𝑅 ∈𝐹𝑁𝑅(𝑋×𝑍 ) to the one defined
𝜆,𝜌
by
𝛼,𝛽
𝑃 ° 𝑅 = {〈(𝑥,𝑧),𝑇 𝛼,𝛽 (𝑥,𝑧),𝐼 𝛼,𝛽 (𝑥,𝑧),𝐹 𝛼,𝛽 (𝑥,𝑧)〉/𝑥 ∈ 𝑋,𝑧 ∈ 𝑍}
𝜆,𝜌 𝑃 ° 𝑅 𝑃 ° 𝑅 𝑃 ° 𝑅
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌
Where,𝑇 (𝑥,𝑧) = 𝛼{𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]},𝐼 (𝑥,𝑧) = 𝛼{𝛽[𝐼 (𝑥,𝑦),𝐼 (𝑦,𝑧)]} ,
𝛼,𝛽 𝑦 𝑅 𝑃 𝛼,𝛽 𝑦 𝑅 𝑃
𝑃 ° 𝑅 𝑃 ° 𝑅
𝜆,𝜌 𝜆,𝜌
𝐹 (𝑥,𝑧) = 𝜆{𝜌[𝐹 (𝑥,𝑦),𝐹 (𝑦,𝑧)]}
𝛼,𝛽 𝑦 𝑅 𝑃
𝑃 ° 𝑅
𝜆,𝜌
Whenever 0 ≤ 𝑇 (𝑥,𝑧)+𝐼 (𝑥,𝑧)+𝐹 (𝑥,𝑧) ≤ 3 ∀(𝑥,𝑧) ∈ 𝑋×𝑍 .
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
𝑃 ° 𝑅 𝑃 ° 𝑅 𝑃 ° 𝑅
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌
The choice of the t-norms and t-conorms 𝛼,𝛽,𝜆,𝜌 in the previous definition, is evidently
conditioned by the fulfilment of
0 ≤ 𝑇 (𝑥,𝑧)+𝐼 (𝑥,𝑧)+𝐹 (𝑥,𝑧) ≤ 3 ∀(𝑥,𝑧) ∈ 𝑋×𝑍
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
𝑃 ° 𝑅 𝑃 ° 𝑅 𝑃 ° 𝑅
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌

Theorem 3.6:

For each, ∈ 𝐹𝑁𝑅(𝑋×𝑌) , 𝑃 ∈ 𝐹𝑁𝑅(𝑌×𝑍) and 𝛼,𝛽,𝜆,𝜌 any t-norms or t-conorms
−1
𝛼,𝛽 𝛼,𝛽
(𝑃 ° 𝑅) = 𝑅−1 ° 𝑃−1 is fulfilled.
𝜆,𝜌 𝜆,𝜌

Proof:

𝑇 (𝑧,𝑥) = 𝑇 (𝑥,𝑧) = 𝛼{𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]} = 𝛼{𝛽[𝑇 (𝑦,𝑥),𝑇 (𝑧,𝑦)]}
𝛼,𝛽 −1 𝛼,𝛽 𝑦 𝑅 𝑃 𝑦 𝑅−1 𝑃−1
(𝑃 ° 𝑅) 𝑃 ° 𝑅
𝜆,𝜌 𝜆,𝜌
= 𝛼{𝛽[𝑇 (𝑧,𝑦),𝑇 (𝑦,𝑥)]} = 𝑇 (𝑧,𝑥). Similarly 𝐼 (𝑧,𝑥) = 𝐼 (𝑧,𝑥),
𝑦 𝑃−1 𝑅−1 𝑅−1 𝛼°,𝛽 𝑃−1 (𝑃𝛼°,𝛽𝑅)−1 𝑅−1 𝛼°,𝛽 𝑃−1
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌
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𝐹 (𝑧,𝑥) = 𝐹 (𝑥,𝑧) = 𝜆{𝜌[𝐹 (𝑥,𝑦),𝐹 (𝑦,𝑧)]} = 𝜆{𝜌[𝐹 (𝑦,𝑥),𝐹 (𝑧,𝑦)]}
𝛼,𝛽 −1 𝛼,𝛽 𝑦 𝑅 𝑃 𝑦 𝑅−1 𝑃−1
(𝑃 ° 𝑅) 𝑃 ° 𝑅
𝜆,𝜌 𝜆,𝜌
= 𝜆{𝜌[𝐹 (𝑧,𝑦),𝐹 (𝑦,𝑥)]} = 𝐹 (𝑧,𝑥) ∀(𝑧,𝑥) ∈ 𝑍×𝑋.
𝑦 𝑃−1 𝑅−1 𝛼,𝛽
𝑅−1 ° 𝑃−1
𝜆,𝜌
−1
𝛼,𝛽 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 𝑃 ≤ 𝑃 , then 𝑃 ∘ 𝑅 ≤ 𝑃 ∘ 𝑅, for every R ∈ FNR
1 2 1 2
𝜆,𝜌 𝜆,𝜌
𝛼,𝛽 𝛼,𝛽
If 𝑅 ≤ 𝑅 , then 𝑃 ∘ 𝑅 ≤ 𝑃 ∘ 𝑅 , for every P ∈ FNR
1 2 1 2
𝜆,𝜌 𝜆,𝜌
𝛼,𝛽 𝛼,𝛽
If 𝑃 ≼ 𝑃 , then 𝑃 ∘ 𝑅 ≼ 𝑃 ∘ 𝑅, for every R ∈ FNR
1 2 1 2
𝜆,𝜌 𝜆,𝜌
𝛼,𝛽 𝛼,𝛽
If 𝑅 ≼ 𝑅 , then 𝑃 ∘ 𝑅 ≼ 𝑃 ∘ 𝑅 , for every P ∈ FNR
1 2 1 2
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:

𝑃 ≤ 𝑃 then 𝑇 (𝑦,𝑧) ≤ 𝑇 (𝑦,𝑧),𝐼 (𝑦,𝑧) ≤ 𝐼 (𝑦,𝑧) 𝑎𝑛𝑑 𝐹 (𝑦,𝑧) ≥ 𝐹 (𝑦,𝑧)
1 2 𝑃1 𝑃2 𝑃1 𝑃2 𝑃1 𝑃2
𝛼 𝛼
𝑇 (𝑥,𝑧) = {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]} ≤ {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]} = 𝑇 (𝑥,𝑧).
𝛼,𝛽 𝑦 𝑅 𝑃1 𝑦 𝑅 𝑃2 𝛼,𝛽
𝑃1 ° 𝑅 𝑃2 ° 𝑅
𝜆,𝜌 𝜆,𝜌
Similarly 𝐼 (𝑥,𝑧) ≤ 𝐼 (𝑥,𝑧)
𝛼,𝛽 𝛼,𝛽
𝑃1 ° 𝑅 𝑃2 ° 𝑅
𝜆,𝜌 𝜆,𝜌
𝜆 𝜆
𝐹 (𝑥,𝑧) = {𝜌[𝐹 (𝑥,𝑦),𝐹 (𝑦,𝑧)]} ≥ {𝜌[𝐹 (𝑥,𝑦),𝐹 (𝑦,𝑧)]} = 𝐹 (𝑥,𝑧)
𝛼,𝛽 𝑦 𝑅 𝑃1 𝑦 𝑅 𝑃2 𝛼,𝛽
𝑃1 ° 𝑅 𝑃2 ° 𝑅
𝜆,𝜌 𝜆,𝜌
𝛼,𝛽 𝛼,𝛽
Therefore 𝑃 ∘ 𝑅 ≤ 𝑃 ∘ 𝑅
1 2
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
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Proof:

Starting from the points (vi), (vii), (viii) of the theorem 3.4, we get
𝛼,𝛽 𝛼,𝛽

(𝑅∨𝑃) ∘ 𝑄 ≥ 𝑅 ∘ 𝑄
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
𝑅 ∨ 𝑃 ≥ 𝑅 𝜆,𝜌 𝜆,𝜌
{ ⇒ ⇒ (𝑅 ∨𝑃) ∘ 𝑄 ≥ 𝑅 ∘ 𝑄 ∨ 𝑃 ∘ 𝑄
𝑅 ∨ 𝑃 ≥ 𝑃 𝛼,𝛽 𝛼,𝛽
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌
(𝑅∨𝑃) ∘ 𝑄 ≥ 𝑃 ∘ 𝑄
{ 𝜆,𝜌 𝜆,𝜌
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 (𝑌× Z), Q ∈ 𝐹𝑁𝑅(𝑋×𝑌) ,𝛼 𝑎𝑛𝑑 𝜆 not null t-norms or
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
t -conorms. Then (𝑅∨𝑃) ∘ 𝑄 = 𝑅 ∘ 𝑄 ∨ 𝑃 ∘ 𝑄 if and only if 𝛼 = ∨𝑎𝑛𝑑 𝜆 = ∧.
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌
Proof:
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
𝛼
Let(𝑅 ∨𝑃) ∘ 𝑄 = 𝑅 ∘ 𝑄 ∨ 𝑃 ∘ 𝑄 .𝑇 (𝑥,𝑥) = {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)∨𝑇 (𝑦,𝑧)]}
𝛼,𝛽 𝑄 𝑅 𝑃
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌 (𝑅∨𝑃) ∘ 𝑄 𝑦
𝜆,𝜌
𝛼
= {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)] ∨ 𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]}.Because of the hypothesis of the theorem the
𝑄 𝑅 𝑄 𝑃
𝑦
𝛼 𝛼
result is = {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]}∨ {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]}.
𝑄 𝑅 𝑄 𝑃
𝑦 𝑦
Let {𝑎 } be any two finite family of numbers belonging to the interval [0, 1]
𝑦
𝑦∈𝑌
If 𝛽 is t-norm, we define (for x, z fixed and for every y).
𝑇 (𝑥,𝑦) = 1,𝑇 (𝑦,𝑧) = 𝑎 ,𝑇 (𝑦,𝑧) = 𝑏 .Then it is known that 𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)] = 𝑎
𝑄 𝑅 𝑦 𝑃 𝑦 𝑄 𝑅 𝑦
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)] = 𝑏 .Besides as (𝑅∨𝑃) ∘ 𝑄 = (𝑅 ∘ 𝑄) ∨ (𝑃 ∘ 𝑄) it is verified
𝑄 𝑃 𝑦
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌
for every R, P and Q by means of hypothesis we have 𝛼(𝑎 ∨ 𝑏 ) = 𝛼(𝑎 )∨ 𝛼(𝑏 ) and this
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
condition verified for every {𝑎 } , {𝑏 } .We have proved in the theorem 2.0, that is
𝑦 𝑦
𝑦∈𝑌 𝑦∈𝑌
equivalent to 𝛼 = ∨
(ii) If 𝛽 is t-conorm, we define the degree of truthfulness of R, P and Q as follows:
𝑇 (𝑥,𝑦) = 0,𝑇 (𝑦,𝑧) = 𝑎 ,𝑇 (𝑦,𝑧) = 𝑏 and with the same proceeding we conclude that
𝑄 𝑅 𝑦 𝑃 𝑦
verifying 𝛼(𝑎 ∨ 𝑏 ) = 𝛼(𝑎 )∨ 𝛼(𝑏 ) as we have seen in the theorem 2.0, if and only if
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
𝛼 = ∨.
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 𝛼 = ∨𝑎𝑛𝑑 𝜆 = ∧,𝛽 𝑎𝑛𝑑 𝜌 be any t-
∨
norms and t-conorms 𝑇 (𝑥,𝑧) = {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧) ∨ 𝑇 (𝑦,𝑧)]}
𝛼,𝛽 𝑄 𝑅 𝑃
(𝑅∨𝑃) ∘ 𝑄 𝑦
𝜆,𝜌
∨
= {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)] ∨ 𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]}
𝑄 𝑅 𝑄 𝑃
𝑦
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Using the associative property of the t-conorms, we have
∨ ∨
= {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]} ∨ {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]} = 𝑇 (𝑥,𝑧) ∨ 𝑇 (𝑥,𝑧)
𝑄 𝑅 𝑄 𝑃 𝛼,𝛽 𝛼,𝛽
𝑦 𝑦 𝑅 ∘ 𝑄 𝑃 ∘ 𝑄
𝜆,𝜌 𝜆,𝜌
∀ (𝑥,𝑧) ∈ 𝑋×𝑍 .Similarly 𝐼 (𝑥,𝑧) = 𝐼 (𝑥,𝑧) ∨ 𝐼 (𝑥,𝑧) and
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
(𝑅∨𝑃) ∘ 𝑄 𝑅 ∘ 𝑄 𝑃 ∘ 𝑄
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌
𝐹 (𝑥,𝑧) = 𝐹 (𝑥,𝑧) ∧ 𝐹 (𝑥,𝑧)
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
(𝑅∨𝑃) ∘ 𝑄 𝑅 ∘ 𝑄 𝑃 ∘ 𝑄
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌

Theorem 3.10:

For every 𝛼,𝛽,𝜆,𝜌 any t-norms or t-conorms and 𝑅,𝑃 ∈ 𝐹𝑁𝑅(𝑌×𝑍) , 𝑄 ∈ 𝐹𝑁𝑅(𝑋×𝑌), it is
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
verified that (𝑅∧𝑃) ∘ 𝑄 ≤ (𝑅 ∘ 𝑄) ∧ (𝑃 ∘ 𝑄)
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌

Proof:

Analogous to the one made in the theorem 3.7.

Theorem 3.11:

Let R, P be the two elements of FNR (𝑌× Z), Q ∈ 𝐹𝑁𝑅(𝑋×𝑌)
𝛼 different from the null t-norm 𝑎𝑛𝑑 𝜆 different from the null t-conorm. Then
𝛼,𝛽 𝛼,𝛽 𝛼,𝛽
(𝑅∧𝑃) ∘ 𝑄 = (𝑅 ∘ 𝑄) ∧ (𝑃 ∘ 𝑄) if and only if 𝛼 = ∧𝑎𝑛𝑑 𝜆 = ∨.
𝜆,𝜌 𝜆,𝜌 𝜆,𝜌

Proof:

The proof follows by theorem 3.9.
From the analysis of the previous theorem it is deduced that the choice of 𝛼,𝛽,𝜆 𝑎𝑛𝑑 𝜌 t-norms
or t-conorms will depend on the problem traced on each case. However the distributive equalities
will demand the choice of ∨𝑎𝑛𝑑 ∧𝑓𝑜𝑟 𝛼 𝑎𝑛𝑑 𝜆 𝑜𝑟 𝜆 𝑎𝑛𝑑 𝛼 respectively.

Theorem 3.12:

Let Q ∈ 𝐹𝑁𝑅(𝑋×𝑌) ,P ∈ 𝐹𝑁𝑅(𝑌×𝑍),R ∈ 𝐹𝑁𝑅(𝑍×𝑍) 𝛽 𝑎𝑛𝑑 𝜌 any t-norms or t-conorms.
∨,𝛽 ∨,𝛽 ∨,𝛽 ∨,𝛽
If 𝛼 = ∨𝑎𝑛𝑑 𝜆 = ∧ then (𝑅 ∘ 𝑃) ∘ 𝑄 = 𝑅 ∘ (𝑃 ∘ 𝑄)
∧,𝜌 ∧,𝜌 ∧,𝜌 ∧,𝜌

Proof:
Let 𝛽 be associative ∨ ∨𝑎 = ∨ ∨𝑎 , 𝛽(𝑎,⋁𝑏 ) = ⋁(𝛽(𝑎,𝑏 )),𝛽( ⋁𝑎 ,𝑏) = ⋁(𝛽(𝑎 𝑏))
𝑖 𝑗 𝑖,𝑗 𝑗 𝑖 𝑖,𝑗 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖,
and the same properties are applied to ∧.𝑇 (𝑥,𝑧) = ∨ {𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧) ]}
∨,𝛽 ∨,𝛽 𝑦 𝑄 ∨,𝛽
(𝑅 ∘ 𝑄) ∘ 𝑄 𝑅 ∘ 𝑃
∧,𝜌 ∧,𝜌 ∧,𝜌
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∨
= ∨ {𝛽[𝑇 (𝑥,𝑦), {𝛽(𝑇 (𝑦,𝑡),(𝑇 (𝑡,𝑧)) )} ]} = ∨ {∨ {𝛽[𝑇 (𝑥,𝑦), 𝛽(𝑇 (𝑦,𝑡),𝑇 (𝑡,𝑧)) ]}}
𝑦 𝑄 𝑃 𝑅 𝑦 𝑡 𝑄 𝑃 𝑅
𝑡
= ∨ ∨ {𝛽[𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑡),𝑇 (𝑡,𝑧)]]} = ∨ {𝛽[∨ 𝛽 [𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑡),𝑇 (𝑡,𝑧) ]]}
𝑡 𝑦 𝑄 𝑃 𝑅 𝑡 𝑦 𝑄 𝑃 𝑅
= ∨ {𝛽[𝑇 (𝑥,𝑡),𝑇 (𝑡,𝑧)]} = 𝑇 (𝑥,𝑧) ∀ (𝑥,𝑧) ∈ 𝑋×𝑍
𝑡 ∨,𝛽 𝑅 ∨,𝛽 ∨,𝛽
𝑃 ∘ 𝑄 𝑅 ∘ (𝑃 ∘ 𝑄)
∧,𝜌 ∧,𝜌 ∧,𝜌
The equality 𝐼 (𝑥,𝑧) = 𝐼 (𝑥,𝑧) 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 (𝑥,𝑧) ∈ 𝑋×𝑍 and
∨,𝛽 ∨,𝛽 ∨,𝛽 ∨,𝛽
(𝑅 ∘ 𝑃) ∘ 𝑄 𝑅 ∘ (𝑃 ∘ 𝑄)
∧,𝜌 ∧,𝜌 ∧,𝜌 ∧,𝜌
𝐹 (𝑥,𝑧) = 𝐹 (𝑥,𝑧) ∀ (𝑥,𝑧) ∈ 𝑋×𝑍 corresponding to the indeterminacy and
∨,𝛽 ∨,𝛽 ∨,𝛽 ∨,𝛽
(𝑅 ∘ 𝑃) ∘ 𝑄 𝑅 ∘ (𝑃 ∘ 𝑄)
∧,𝜌 ∧,𝜌 ∧,𝜌 ∧,𝜌
falsity are proved in a similar way.

Note:
𝛼,𝛽
It is convenient to state that in the notation used for composition 𝑃 ° 𝑅 , the symbols 𝛼 and 𝛽
𝜆,𝜌
placed up are applied to the truth value and indeterministic value and the symbols 𝜆 and 𝜌 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 ∈ 𝐹𝑁𝑅(𝑋×𝑋) is called the relation of identity if ∀ (𝑥,𝑦) ∈ 𝑋×𝑋
1 𝑖𝑓 𝑥 = 𝑦 1 𝑖𝑓 𝑥 = 𝑦 0 𝑖𝑓 𝑥 = 𝑦
𝑇 (𝑥,𝑦) = { , 𝐼 (𝑥,𝑦) = { , 𝐹 (𝑥,𝑦) = {
∆ 0 𝑖𝑓 𝑥 ≠ 𝑦 ∆ 0 𝑖𝑓 𝑥 ≠ 𝑦 ∆ 1 𝑖𝑓 𝑥 ≠ 𝑦

The complementary relation ∆𝑐= ∇ is defined by

0 𝑖𝑓 𝑥 = 𝑦 0 𝑖𝑓 𝑥 = 𝑦 1 𝑖𝑓 𝑥 = 𝑦
𝑇 (𝑥,𝑦) = { , 𝐼 (𝑥,𝑦) = { , 𝐹 (𝑥,𝑦) = { .
∇ 1 𝑖𝑓 𝑥 ≠ 𝑦 ∇ 1 𝑖𝑓 𝑥 ≠ 𝑦 ∇ 0 𝑖𝑓 𝑥 ≠ 𝑦
It is evident that ∆ = ∆−1 and ∇ = ∇−1

Theorem 4.2:

Let 𝛼,𝛽,𝜆,𝜌 be t-norms and t-conorms and 𝑅 ∈ 𝐹𝑁𝑅(𝑋×𝑋)
𝛼,𝛽 𝛼,𝛽
𝑅 ∘ ∆ = ∆ ∘ 𝑅 = 𝑅 if and only if 𝛼 is t-conorm , 𝛽 is t-norm , 𝜆 is t-norm and 𝜌 is t-
𝜆,𝜌 𝜆,𝜌
conorm.
𝜆,𝜌 𝜆,𝜌
𝑅 ∘ ∇ = ∇ ∘ 𝑅 = 𝑅 if and only if 𝛼 is t-conorm , 𝛽 is t-norm , 𝜆 is t-norm and 𝜌 is t-
𝛼,𝛽 𝛼,𝛽
conorm.

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[Arockiarani et. al., Vol.4 (Iss.2): February, 2016] ISSN- 2350-0530(O) ISSN- 2394-3629(P)
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Proof:

𝑇 (𝑥,𝑧) = 𝛼{𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)]} = 𝛼{𝛽[𝑇 (𝑥,𝑥),𝑇 (𝑥,𝑧)],𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑧)] }
𝛼,𝛽 𝑦 ∆ 𝑅 𝑦≠𝑥 ∆ 𝑅 ∆ 𝑅
𝑅 ∘ ∆
𝜆,𝜌
= 𝛼{𝛽[1,𝑇 (𝑥,𝑧)],𝛽[0,𝑇 (𝑦,𝑧)] } = 𝛼{[𝑇 (𝑥,𝑧),0] } = 𝑇 (𝑥,𝑧) ,(𝑥,𝑧) ∈ 𝑋×𝑋
𝑦≠𝑥 𝑅 𝑅 𝑦≠𝑥 𝑅 𝑅
Similarly, 𝐼 (𝑥,𝑧) = 𝐼 (𝑥,𝑧) ,(𝑥,𝑧) ∈ 𝑋×𝑋
𝛼,𝛽 𝑅
𝑅 ∘ ∆
𝜆,𝜌
𝐹 (𝑥,𝑧) = 𝜆{𝜌[𝐹 (𝑥,𝑦),𝐹 (𝑦,𝑧)]} = 𝜆{𝜌[𝐹 (𝑥,𝑥),𝐹 (𝑥,𝑧)],𝜌[𝐹 (𝑥,𝑦),𝐹 (𝑦,𝑧)] }
𝛼,𝛽 𝑦 ∆ 𝑅 𝑦≠𝑥 ∆ 𝑅 ∆ 𝑅
𝑅 ∘ ∆
𝜆,𝜌
= 𝜆{𝜌[0,𝐹 (𝑥,𝑧)],𝜌[1,𝐹 (𝑦,𝑧)] } = 𝜆{[𝐹 (𝑥,𝑧),1] } = 𝐹 (𝑥,𝑧) ,(𝑥,𝑧) ∈ 𝑋×𝑋
𝑦≠𝑥 𝑅 𝑅 𝑦≠𝑥 𝑅 𝑅
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 𝑥 = 𝑧 then
∆
𝑇 (𝑥,𝑥) = 𝛼{𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑥)]} = 𝛼{𝛽[𝑇 (𝑥,𝑥),𝑇 (𝑥,𝑥)],𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑥)] }
𝛼,𝛽 𝑦 ∆ ∆ 𝑦≠𝑥 ∆ ∆ ∆ ∆
∆ ∘ ∆
𝜆,𝜌
= 𝛼{𝛽[1,1],𝛽[0,0] } = 𝛼{[1,0] } = 0 ≠ 𝑇 (𝑥,𝑥) = 1 ∀𝑥 ∈ 𝑋 Which is a contradiction.
𝑦≠𝑥 𝑦≠𝑥 ∆

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

For every (𝑥,𝑧) ∈ 𝑋×𝑋, 𝑇 (𝑥,𝑧) ≠ 𝑇 (𝑥,𝑧) if 𝑥 ≠ 𝑧
𝛼,𝛽 ∆
∆ ∘ ∆
𝜆,𝜌
Case (iii):
1 𝑖𝑓 𝑥 = 𝑦
Let 𝛼 be t-norm and 𝛽 be t-conorm. Taking 𝑅 in the following way 𝑇 (𝑥,𝑦) = { .
𝑅 ≠ 1 𝑖𝑓 𝑥 ≠ 𝑦
By means of hypothesis 𝑇 (𝑥,𝑥) = 𝑇 (𝑥,𝑥) = 1 ∀ 𝑥 ∈ 𝑋 have to be fulfilled.
𝛼,𝛽 𝑅
𝑅 ∘ ∆
𝜆,𝜌
𝑇 (𝑥,𝑥) = 𝛼{𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑥)]} = 𝛼{𝛽[𝑇 (𝑥,𝑥),𝑇 (𝑥,𝑥)],𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑥)] }
𝛼,𝛽 𝑦 ∆ 𝑅 𝑦≠𝑥 ∆ 𝑅 ∆ 𝑅
𝑅 ∘ ∆
𝜆,𝜌
= 𝛼{𝛽[1,1],𝛽[0,𝑇 (𝑦,𝑥)] } = 𝛼{1,𝑇 (𝑦,𝑥) } = 𝑇 (𝑦,𝑥) ≠ 1 = 𝑇 (𝑥,𝑥) which is not true.
𝑦≠𝑥 𝑅 𝑦≠𝑥 𝑅 𝑅 𝑅

Http://www.granthaalayah.com ©International Journal of Research - GRANTHAALAYAH [17-30]
[Arockiarani et. al., Vol.4 (Iss.2): February, 2016] ISSN- 2350-0530(O) ISSN- 2394-3629(P)
Impact Factor: 2.035 (I2OR)
Case (iv):
1 𝑖𝑓 𝑥 = 𝑦
Let 𝛼 be t-conorm and 𝛽 be t-norm .Taking 𝑅 in the following way 𝑇 (𝑥,𝑦) = {
𝑅 ≠ 1 𝑖𝑓 𝑥 ≠ 𝑦
by means of hypothesis 𝑇 (𝑥,𝑥) = 𝑇 (𝑥,𝑥) = 1 ∀ 𝑥 ∈ 𝑋 have to be fulfilled.
𝛼,𝛽 𝑅
𝑅 ∘ ∆
𝜆,𝜌
𝑇 (𝑥,𝑥) = 𝛼{𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑥)]} = 𝛼{𝛽[𝑇 (𝑥,𝑥),𝑇 (𝑥,𝑥)],𝛽[𝑇 (𝑥,𝑦),𝑇 (𝑦,𝑥)] }
𝛼,𝛽 𝑦 ∆ 𝑅 𝑦≠𝑥 ∆ 𝑅 ∆ 𝑅
𝑅 ∘ ∆
𝜆,𝜌
= 𝛼{𝛽[1,1],𝛽[0,𝑇 (𝑦,𝑥)] } = 𝛼{1,0 } = 1 = 𝑇 (𝑥,𝑥).
𝑦≠𝑥 𝑅 𝑦≠𝑥 𝑅

Hence it is proved that 𝛼 is t-conorm and 𝛽 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 𝜆 is t-norm and 𝜌 is t-conorm.

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

Definition 4.3:

The relation 𝑅 ∈ 𝐹𝑁𝑅(𝑋×𝑌) is called
Reflexive if for every 𝑥 ∈ 𝑋 ,𝑇 (𝑥,𝑥) = 1 ,𝐼 (𝑥,𝑥) = 1,𝐹 (𝑥,𝑥) = 0
𝑅 𝑅 𝑅
Anti- reflexive if for every 𝑥 ∈ 𝑋 ,𝑇 (𝑥,𝑥) = 0 ,𝐼 (𝑥,𝑥) = 0,𝐹 (𝑥,𝑥) = 1
𝑅 𝑅 𝑅
(i.e,) The relation 𝑅 is called anti-reflexive if its complement 𝑅𝑐 is reflexive.

Theorem 4.4:

For every 𝑅 ∈ 𝐹𝑁𝑅(𝑋×𝑌), it is verified that
If 𝑅 is reflexive then ∆ ≤ 𝑅 (ii) If 𝑅 is anti-reflexive then ∆ ≥ 𝑅

Proof:

It is the consequence of the definition 4.1 and 4.3.

Theorem 4.5:

For 𝛼 t-conorm, 𝛽 t-norm, 𝜆 t-norm and 𝜌 t-conorm it is verified that
𝛼,𝛽
If 𝑅 ∈ 𝐹𝑁𝑅(𝑋×𝑌) is reflexive, then 𝑅 ≤ 𝑅 ∘ 𝑅
𝜆,𝜌
𝜆,𝜌
If 𝑅 ∈ 𝐹𝑁𝑅(𝑋×𝑌) is anti-reflexive, then 𝑅 ≥ 𝑅 ∘ 𝑅
𝛼,𝛽

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