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To prove equivalence relation

  1. May 25, 2009 #1
    1. The problem statement, all variables and given/known data

    We have an equivalence relation such that
    A <-> B.

    Prove that the equivalence relation is true.

    3. The attempt at a solution

    Let
    P: A -> B
    Q: B -> A

    Let's prove the relation by contradiction.
    Assume
    [tex]\neg A -> \neg B [/tex]

    The previous assumption is the same as Q. Thus, we have a contradiction, since
    it is impossible that both of the following Q and [tex] \neg Q[/tex]
    are true at the same time, where

    Q: B -> A and
    [tex]\neg Q: \neg A -> \neg B [/tex] which is the same as B -> A.

    Thus, the equivalence relation is true between A and B.
     
  2. jcsd
  3. May 25, 2009 #2
    Is A <-> B given or is that what you're trying to prove?
     
  4. May 25, 2009 #3
    Are you trying to show that <-> is an equivalence relation on propositions?

    If that is the case, you have to show the following three things:

    A <-> A for all A.
    if A <-> B then B <-> A.
    if A <-> B AND B <-> C then A <-> C.

    How should you show these? I'd probably use truth tables and the definition if <-> in terms of -> and "AND".
     
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