Looking at the fuzzy set A, the membership grades are _A(a) = 0.3, _A(b) = 0.4, _A(c) = 0.5, _A(d) = 0.8, _A(e) = 1.0, and _A(f) = 0.1. The only element with membership grade _A(x) = 1 is e. Therefore, the core of A is e.

Looking at the fuzzy set A, the membership grades are _A(a) = 0.3, _A(b) = 0.4, _A(c) = 0.5, _A(d) = 0.8, _A(e) = 1.0, and _A(f) = 0.1. The only element with membership grade _A(x) = 1 is e. Therefore, the core of A is e.

1) If $A = \frac{0.3}{a} + \frac{0.4}{b} + \frac{0.5}{c} + \frac{0.8}{d} + \frac{1.0}{e} + \frac{0.1}{f}$ and $B = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ be fuzzy sets of $X = \{a, b, c, d, e, f\}$ , then a) find core of $A$ . b) Find support of $B$ .

Answer: a) $\text{core}(A) = \{e\}$ , b) $\text{supp}(B) = \{a, b, c\}$

Step-by-step solution

Step 1: Recall definitions of core and support of a fuzzy set

In fuzzy set theory, a fuzzy set is represented as a collection of elements with their corresponding membership values $\mu(x)$ . The core of a fuzzy set $A$ is the crisp set of all elements in the universe $X$ whose membership grade is exactly $1$ . The support of a fuzzy set $B$ is the crisp set of all elements in $X$ whose membership grade is strictly greater than $0$ .

Step 2: Find the core of fuzzy set A

Looking at the fuzzy set $A$ , the membership grades are $\mu_A(a) = 0.3$ , $\mu_A(b) = 0.4$ , $\mu_A(c) = 0.5$ , $\mu_A(d) = 0.8$ , $\mu_A(e) = 1.0$ , and $\mu_A(f) = 0.1$ . The only element with membership grade $\mu_A(x) = 1$ is $e$ . Therefore, the core of $A$ is $\{e\}$ .

Step 3: Find the support of fuzzy set B

The fuzzy set $B$ is given as $B = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ , which means $\mu_B(a) = 1$ , $\mu_B(b) = 1$ , $\mu_B(c) = 1$ , and for elements d, e, f, the membership grade is $0$ . The support consists of all elements where $\mu_B(x) > 0$ , which gives $\{a, b, c\}$ .

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