Fuzzy Set Union with Universe of Discourse X = {2,4,6,…,20} | Query Point Official

Fuzzy Set Union on a Given Universe of Discourse

Problem Overview

Let X = {2, 4, 6, …, 20} be the universe of discourse. Two fuzzy sets are defined on X:

`A^~1 = {(2,0.5), (4,0.9), …, (20,0.8)}`

`B^~1 = {(2,0.5), (4,0.5), …, (20,0.4)}`

The task is to determine the union of the fuzzy sets: `A^~1 ∪ B^~1`.

Step 1: Understanding Fuzzy Sets

A fuzzy set on a universe X assigns a membership value between 0 and 1 to each element in X. Higher values indicate stronger membership.

Step 2: Compute Fuzzy Set Union

In fuzzy set theory, the union of two fuzzy sets is computed by taking the maximum membership value across sets for each element in X.

So for every x in X:

• `μ_{A∪B}(x) = max( μ_A(x), μ_B(x) )`

Step 3: Apply to Given Sets

Using the provided membership values:

`A^~1 ∪ B^~1 = {(2,0.5), (4,0.5), (6,0.8), (8,0.8), (10,0.7), (12,0.8), (14,0.6), (16,1.0), (18,0.8), (20,0.6)}`

Explanation of the Result

For each element of X, we took the higher of the two membership values from `A^~1` and `B^~1`. This gives the correct fuzzy union according to standard fuzzy set rules.

FAQ

What is the universe of discourse?

The universe of discourse is the set of all elements over which fuzzy sets or predicates are defined — here, the even numbers from 2 to 20.

How do we compute a fuzzy union?

By taking the maximum membership value for each element across the sets.

Why is fuzzy logic used?

Fuzzy logic handles uncertainty by assigning degrees of membership rather than crisp true/false values.

Related Problems

See Mathematics Notes & MCQs for more set theory and logic problems.