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  1. #1
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    Equivalence Relation ~

    I'm having trouble with these types of problems. Any help would be greatly appreciated .

    Let  A_1 = \{0,1\}, A_2 = \{1,2\}, A_3 = \{3,4\} be subsets of S = \{0,1,2,3,4\}. Explain why the relation ~ on S defined by x~y if and only if   x,y \in A_i for some  i \in \{1,2,3\} is NOT an equivalence relation on S.
    I know that in order ~ to be an equivalence relation, ~ must be:
    1) (reflexive) x~ x for all x in X.
    2) (symmetric) If x~ y then y~ x.
    3) (transitive) If x~ y and y~ z, then x~ z.
    But, I just can't find a counter example for which it doesn't satisfy one or more of 1,2,3.
    Last edited by shinn; April 3rd 2009 at 08:43 PM.
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  2. #2
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    Equivalence Relation

    Hello shinn

    Let be subsets of . Explain why the relation ~ on S defined by x~y if and only if for some is NOT an equivalence relation on S.

    I know that in order ~ to be an equivalence relation, ~ must be:
    1) (reflexive) x~ x for all x in X.
    2) (symmetric) If x~ y then y~ x.
    3) (transitive) If x~ y and y~ z, then x~ z.
    But, I just can't find a counter example for which it doesn't satisfy one or more of 1,2,3.
    (1) and (2) are both satisfied. So what about (3)? 0\sim 1 because 0, 1 \in A_1 and 1\sim 2 because 1, 2 \in A_2. But 0\sim 2?

    Grandad
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