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Proving A != B if A -> B and B -> A and A and B are Fuzzy sets

  1. Oct 15, 2008 #1
    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.
  2. jcsd
  3. Oct 15, 2008 #2


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    Staff Emeritus
    Science Advisor

    Exactly what definition of "fuzzy sets" are you using? There are several equivalent definitions and exactly how you prove something like this depends upon which you are using.

    Also note that your title, "A != B if A -> B and B -> A" is NOT the same as "A [itex]\subseteq[/itex] and BA [itex]\subseteq[/itex] A is not sufficient to prove A= B".
  4. Oct 16, 2008 #3
    I am using the definiton of fuzzy sets as defined by Zadeh as being generalizations of classical sets where membership and non-membership is not clearly defined.
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