A Very Short Note on Some of the Fuzzy Operations

Bikram Baro^1* and Sahalad Borgoyary²

^1*Research Scholar, Department of Mathematics, Central Institute of Technology, Kokrajhar, Assam (India)
²Department of Mathematics, Central Institute of Technology, Kokrajhar, Assam (India)

Corresponding Author: barobikram@gmail.com

Abstract

This article reviews commonly used operations of fuzzy sets and compares them with classical (crisp) set operations. It highlights similarities, differences, and analogues through clear definitions, theorems, and illustrative examples. Special aspects of various fuzzy set operations are discussed, including Baruah’s newly proposed definitions. Rather than proposing a new theory, the article offers conceptual insights into operational relationships between fuzzy and crisp sets.

Keywords: Fuzzy sets, fuzzy operations, Boolean algebra, classical set.

Introduction

The four fundamental algebraic operations are addition, subtraction, multiplication, and division. In fuzzy set theory, analogous operations include complement, union, intersection, algebraic sum, and algebraic product. This paper presents an overview of well-established applications of fuzzy operations widely used by researchers. Zadeh (1975) introduced additional operations such as bounded sum and bounded difference, which expanded the operational framework of fuzzy sets. Defuzzification is an important process in which a fuzzy output is converted into a single crisp value, generally using the same membership function employed during fuzzification. Common defuzzification techniques include the centre of area (COA), centre of maxima (COM), mean of maxima (MOM), centre of sums (COS), and weighted average (WA) methods. Fuzzy sets also form a unitary commutative semiring under union and intersection operations (Mizumoto, 1981). This study examines fuzzy set operations that satisfy certain algebraic laws and those that violate classical Boolean principles, based on Zadeh’s definitions and later extensions by a study (Baruah, 2013).

Definitions

The fuzzy sets were first introduced by Lotfi Asker Zadeh in 1965 as an extension of crisp set or non-fuzzy set. The fuzzy set can be defined as follows:

If X is the universal set, a fuzzy set A in X is defined by a membership function denoted by μ_A; that is μ_A: X → [0, 1].

The standard fuzzy operations are fuzzy complement, intersection and union. These are defined respectively as:

Ā(x) = 1 − A(x); (A ∩ B)(x) = min(A(x), B(x)) and (A ∪ B)(x) = max(A(x), B(x)) for all x ∈ X.

For a fuzzy set A in X characterized by a membership function f_A(x) for all x in X, Zadeh defined the product, algebraic sum and absolute difference as follows:

I. Algebraic product: The algebraic product of A and B is denoted by AB and is defined in terms of the memberships functions of A and B by

II. Algebraic Sum: The algebraic sum of A and B is denoted by A + B and is defined by

III. Absolute difference: The absolute difference of A and B is denoted by |A − B| and is defined by

IV. Law of Contradiction and Law of Excluded Middle: In fuzzy set theory, proposed by Zadeh, the following two laws does not hold good.
A ∩ A^c = ∅, A ∪ A^c = X where ∅ and X are empty and universal set respectively.

To verify the above, one has to show that min(A(x), A^c(x)) = 0 and max(A(x), A^c(x)) = 1 respectively. Obviously, the above equations violated for any value, A(x) ∈ (0,1). For example, let us consider a fuzzy set A on X = (0, 1, 2, 3) is A = 0.3/0 + 0.6/1 + 0.1/2 + 1.0/3. Then, A^c = 0.7/0 + 0.4/1 + 0.9/2 + 0/3.
Now, max[A(x), A^c(x)] = (A ∪ A^c)(x) = 0.7/0 + 0.6/1 + 0.9/2 + 1/3 ∀ x ∈ X.
min[A(x), A^c(x)] = (A ∩ A^c)(x) = 0.3/0 + 0.4/1 + 0.1/2 + 0/3 ∀ x ∈ X. Therefore, the above two laws are violated.

V. Fuzzy Complement:
According to a study (Baruah H.K, 2013), a fuzzy set A = {x, μ(x), x ∈ Ω} defined in the way as A = {x, 1−μ(x), x ∈ Ω}, so that its complement becomes A^c = {x, 1, μ(x), x ∈ Ω}. From this definition, it is clear that the result obtained by using the existing definition of complement does not logically satisfy. Various important properties are used by all fuzzy complements. These relates to the equilibrium of a fuzzy complement i.e. which is defined as any value a for which c(a) = 1 − a.
Theorem 1.1. Every fuzzy complement has at most one equilibrium.
Proof: Let c be an arbitrary fuzzy complement. An equilibrium of c is a solution of the equation c(t) − t = 0, t ∈ [0, 1]. We can prove this theorem by demonstrating that any equation c(t) − t = a, where a is a real constant, has at most one solution.

Now, let us assume that t₁ and t₂ are two distinct solutions of the equation c(t) − t = a such that t₁ < t₂. Then, c(t₁) − t₁ = a and c(t₂) − t₂ = a; so that we get,
c(t₁) − t₁ = c(t₂) − t₂ .......... (i) Since c is monotonic decreasing, c(t₁) ≥ c(t₂), and because of t₁ < t₂,
c(t₁) − t₁ > c(t₂) − t₂, which contradicts to (i). Hence, the equation c(t) − t = a must have at most one solution.

The algebraic properties of fuzzy sets under the operations of bounded-sum, bounded-difference and bounded-product which were discussed by Zadeh in 1975 is discussed below.

VI. Bounded-sum: Let A and B be two fuzzy sets. Then the bounded-sum of A and B is defined as
A ⊕ B ⇔ μ_A⊕B = 1 ∧ (μ_A + μ_B), where the operation ∧ represents a minimum.
From the above definition, we see that A ⊕ B = B ⊕ A
(A ⊕ B) ⊕ C = A ⊕ (B ⊕ C), A ⊕ ∅ = A, A ⊕ U = U

VII. Bounded-difference: Let A and B be two fuzzy sets. Then the bounded-sum of A and B is defined as
A ⊖ B ⇔ μ_A⊖B = 0 ∨ (μ_A − μ_B), where the operation ∨ represents maximum.
From the above definition, we have,
A ⊖ B ≠ B ⊖ A, (A ⊖ B) ⊖ C ≤ A ⊖ (B ⊖ C), A ⊖ (A ⊖ B) = A ∩ B, A ⊖ ∅ = A, ∅ ⊖ A = ∅,

VIII. Bounded-Product: Let A and B be two fuzzy sets. Then the bounded-product of A and B is defined as A ⊙ B ⇔ μ_A⊙B = 0 ∨ (μ_A + μ_B − 1), where the operation ∨ represents maximum.
From the above definition, we see that
A ⊙ B = B ⊙ A, (A ⊙ B) ⊙ C = A ⊙ (B ⊙ C), A ⊙ ∅ = ∅, A ⊙ U = A, A ⊙ A = A

IX. Absolute difference: For two fuzzy sets A and B, their absolute difference is defined as |A − B| = |f_A − f_B|, and is denoted by |A − B|.

Example 1. Let us consider two fuzzy sets given by A = {0.4/0 + 0.5/1 + 0.6/2 + 0.7/3} and B = {0.3/0 + 0.4/1 + 0.4/2 + 0.3/3}.
Here, we are going to find out algebraic sum, algebraic product, bounded sum and bounded product of the given two fuzzy sets. From the above definition we have, algebraic sum of the fuzzy sets A and B,
f_A+B = f_A + f_B − f_Af_B = {0.58/0 + 0.70/1 + 0.76/2 + 0.79/3} then algebraic product of the fuzzy sets A and B, f_AB = f_Af_B
f_AB = {0.12/0 + 0.20/1 + 0.24/2 + 0.21/3}

bounded-sum of the fuzzy sets A and B, A ⊕ B ⇔ μ_A⊕B = 1 ∧ (μ_A + μ_B) = min[1, (μ_A + μ_B)] = {0.7/0 + 0.9/1 + 1/2 + 1/3}

bounded-difference of the two fuzzy sets A and B, A ⊖ B ⇔ μ_A⊖B = 0 ∨ (μ_A − μ_B) = max[0, (μ_A − μ_B)] = {0.1/0 + 0.1/1 + 0.2/2 + 0.4/3}

Various Types of Operations on Fuzzy Sets

Apart from operations of union and intersection, some important combinations of fuzzy set are discussed (Zadeh, 1965). Algebraic difference of the fuzzy sets A(x) and B(x), for all x ∈ X, is denoted by A(x) − B(x), which is defined as A(x) − B(x) = [(x, μ_A(x)), x ∈ X], where μ_A−B(x) = μ_A(x). Therefore, if A = {(1,0.1), (2,0.5), (3,0.6), (4,0.7)}, B = {(1,0.3), (2,0.4), (3,0.8), (4,0.9)} so that, A(x) − B(x) = {(1,0.1), (2,0.5), (3,0.6), (4,0.7)}.

The disjunctive sum of two fuzzy sets is also a very important operation in fuzzy set theory. If A and B be two fuzzy sets then, A(x) = 1 − A(x), B(x) = 1 − B(x), (A ∩ B)(x) = min(A(x), 1 − B(x)) and (B ∩ A)(x) = min(B(x), 1 − A(x)). Then, the disjunctive sum of two fuzzy sets A and B is defined by (A ⊕ B)(x) = max[min(A(x), 1 − B(x)), min(1 − A(x), B(x))]. For example, if A = {(1,0.3), (2,0.5), (3,0.7), (4,1.0)} and B = {(1,0.6), (2,0.2), (3,0.0), (4,0.3)}, then A = {(1,0.7), (2,0.5), (3,0.3), (4,0.0)} and B = {(1,0.4), (2,0.8), (3,1.0), (4,0.7)}.

Theorem 1.2. The standard fuzzy union is the only idempotent t-conorm.
Proof. We know that, max(a,a) = a for all a ∈ [0,1]. Let us assume that there exists a t-conorm such that t(a,a) = a for all a ∈ [0,1]. Then, for any a, b ∈ [0,1], we have the following two cases: Case 1: a ≥ b: If a ≥ b, then a = i(a,a) ≥ i(a,b) ≥ i(a,0) = a i.e., i(a,b) ≥ a. Therefore, i(a,b) = a = max(a,b)
Case 2: b ≤ a: If a ≤ b, then we get b = i(b,b) ≥ i(a,b) ≥ i(0,b) = b i.e., i(a,b) ≥ b. Hence, i(a,b) = max(a,b) for all a, b ∈ [0,1]. Consequently, from the both cases we have, i(a,b) = max(a,b) for all a, b ∈ [0,1]. In the similar way, we can also show that the standard fuzzy intersection is the only idempotent t-norm. One of the most commonly used operations on fuzzy sets is Cartesian product of fuzzy sets. We all know that the Cartesian product of two non-empty sets A and B is defined as A × B = {(a,b): a ∈ A, b ∈ B}. In general, the Cartesian product of a family of sets {A₁,A₂,...,A_n} is the set of all n-tuples {(a₁,a₂,...,a_n)} such that, a_i ∈ A_i (i=1,2,...,n), written as A₁ × A₂ × ... × A_n = {(a₁,a₂,...,a_n): a_i ∈ A_i for all i}. A fuzzy set A in X characterized by a membership function f_A(x), μ_A(x) or λ_A(x), ..., can be expressed that A = {(x, f_A(x)), x ∈ X}. Then the Cartesian product of the fuzzy sets A₁,A₂,...,A_n in X is expressed as A₁ × A₂ × ... × A_n = {(x₁,x₂,...,x_n): (x_i, f_Ai(x_i)), x_i ∈ X}.

For example, let us consider the set X = {1,2,3} and two fuzzy sets A = {(1,0.2), (2,0.5), (3,0.8)} and B = {(1,0.7), (2,0.4), (4,0.1)}, then the cartesian product of fuzzy sets A and B is expressed as follows: A × B = {((1,1),0.2), ((1,2),0.2), ((1,4),0.1), ((2,1),0.5), ((2,2),0.4), ((2,4),0.1), ((3,1),0.7), ((3,2),0.4), ...}