(adsbygoogle = window.adsbygoogle || []).push({}); Need help proving A != B if A -> B and B -> A and A and B are Fuzzy sets

1. The problem statement, all variables and given/known data

Let X be the universe of discourse where x [tex]\in[/tex] X. Let A and B be non-empty fuzzy sets in X. If A and B are not classical sets then A [tex]\subseteq[/tex] B and B [tex]\subseteq[/tex] A is not sufficient enough to show that A = B.

This is what I'm trying to solve I think I wrote it correctly but if it needs rewording please let me know.

2. Relevant equations

If A [tex]\subseteq[/tex] B then A(x) [tex]\leq[/tex] B(x)

A'(x) = 1 - A(x)

(A [tex]\cap[/tex] B)(x) = min{A(x), B(x)}

3. The attempt at a solution

Proof: Assume A = B. This means that A and B share every element in common. If A = B then the evaluation of (A' [tex]\cap[/tex] B)(x) = min(1-A(x), B(x)) = 0. But min(1-A(x), B(x)) = 0 only when for all x such that, A(x) = B(x) = 1. This means A and B are classical sets but it is established by the hypothesis that A and B are not classical sets. So A [tex]\neq[/tex] B if A [tex]\subseteq[/tex] B and B [tex]\subseteq[/tex] A and A and B are not classical sets.

This is what I reasoned but I'm not feeling confident about this if I'm going about it the right way. If there is anything wrong with it I would really like to know.

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# Homework Help: Proving A != B if A -> B and B -> A and A and B are Fuzzy sets

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