A few more on intuitionistic fuzzy set

Besides the various basic operations for intuitionistic fuzzy sets already available in literature, a new operation is introduced in this paper. Some properties of this operation are discussed and some new relations are also established. At the same time the behavior of modal operators over these operations are rigorously studied. A problem regarding the rank of the students in a certain examination is discussed and the solution is obtained with the help of this new operation.


Introduction
In 1983, Atanassov [1] has done an excellent job by introducing the concept of intuitionistic fuzzy set as an extension of fuzzy set earlier invented by L.A.Zadeh [9] in 1965.Since then many authors and researchers are giving much attention as well as concentration for developing intuitionistic fuzzy sets.In recent past, some results on algebraic laws in intuitionistic fuzzy sets [3,5,7,8] and some basic relation among modal operators [6] are discussed.It is also well known to us that every fuzzy set is intuitionistic fuzzy set but the reverse is not true.But more importantly there exist some operators by which we can transform intuitionistic fuzzy sets into fuzzy sets easily.As discussing the past, present and future of intuitionistic fuzzy sets, Atanassov [4] has remarkably mentioned about the importance of modal operators which are analogous of the modal logic operators 'necessity' and 'possibility'.Here we concentrate deeply upon these operators and try to solve a real life problem related to examination where the percentage of hesitation margin is known well ahead.

Preliminaries
Throughout this paper, intuitionistic fuzzy set and fuzzy set are denoted by IFS and FS respectively.http://www.ispacs.com/journals/jfsva/2016/jfsva-00322/International Scientific Publications and Consulting Services Definition 2.1.[9].Let X be a nonempty set.A fuzzy set A drawn from X is defined as A = {<x, μA(x)>:x∈X}, where μA(x): x→[0,1] is the membership function of the fuzzy set A. Fuzzy set is a collection of objects with graded membership i.e. having degrees of membership.Definition 2.2.[2].Let X be a nonempty set.An intuitionistic fuzzy set A in X is an object having the form A= {<x, μA(x),νA(x)>:x∈X}, where the functions μA(x),νA(x): X→[0,1] define respectively, the degree of membership and degree of non-membership of the element x∈X to the set A, which is a subset of X, and for every element x∈X, 0 ≤ μA(x) + νA(x) ≤1.Furthermore, we have πA(x)= 1-μA(x) -νA(x) called the intuitionistic fuzzy set index or hesitation margin of x in A. πA(x) is the degree of indeterminacy of x∈X to the IFS A and πA(x) ∈[0,1] i.e, πA(x): X →[0,1] and 0 ≤ πA(x) ≤1 for every x∈X.πA(x) expresses the lack of knowledge of whether x belongs to IFS A or not.

Theorem 3.2. Let X be a nonempty set. If A and B be two IFSs drawn from X, then, (a) □ (A-B) = ◊A -◊B (b) ◊ (A-B) = □A -□B
Proof.
Theorem 3.3.Let A and B be two IFSs in a nonempty set X. Then Proof.The proof is straightforward from the definition.

A new operation in IFS Definition 4.1. Let A and B be two IFSs in a nonempty set X. We define the average operation denoted by A ⊜ B as A ⊜ B = < x,½[ μA(x) + μB(x)], ½[ νA(x) + νB(x) >]. Theorem 4.1. For IFSs A and B of the universe X, we have A ⊜ B is IFS.
Proof.
Hence the proof.
Proof.The proof is similar to the above theorem 4.1.

Corollary 4.1. If X be nonempty and A and B be two proper IFSs of X, then, (a) □ (
Proof.Similar to Corollary 3.1.
] > This completes the proof.

Problem
Five students appeared in a Mathematical Olympiad test in which the question paper consists of four different section namely Logical Reasoning, Mathematical Reasoning, Everyday Mathematics and Achiever's Section containing 25 marks each.When the examination is over the answer key is uploaded in the website and the students have confirmed that in the hesitation or uncertain part of their answer 60%, 70%, 60%, 50%, and 40% respectively are correct in each section.Find the rank of the students in the examination.

Solution
Here we use the concept of IFS for representing the marks.The membership degree represents the marks for the correct answer, the non-membership degree represents the marks which is not obtained for incorrect answers and the hesitation degree means the marks which is not confirmed by the students i.e, the students are in hesitation whether the answer is correct or not.Table-1 represents the marks of the students.Since the percentage of the hesitation part is given, we now calculate the membership degree and nonmembership degree using the operator F α,β (A) [by 2.8] and get the table below.

Theorem 4 . 5 .
Let X be nonempty and A,B,C ∈ X are IFSs such that A ⊆ B, then A ⊜ C ⊆A ⊜ C. Proof.Follows from the definition.

Table 1
Student Logical Reasoning Mathematical Reasoning Everyday Mathematics Achiever's Section