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Math Help - is this relation reflexive?

  1. #1
    Member Jskid's Avatar
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    is this relation reflexive?

    The relationship R on Z is defined by (a,a) \in R if and only if a-b is odd. Is this relation reflexive?

    I said no because a-a=0 and 0 is even.

    I'm having trouble getting my head wrapped around reflexive, is this right?
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  2. #2
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    Re: is this relation reflexive?

    Quote Originally Posted by Jskid View Post
    The relationship R on Z is defined by (a,b) \in R if and only if a-b is odd. Is this relation reflexive?
    I said no because a-a=0 and 0 is even.
    You are correct.
    Reflexive simply means that each element is related to itself.
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  3. #3
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    Re: is this relation reflexive?

    the "gold standard" for reflexive relations is: aRb iff a=b (the diagonal relation). a relation S is reflexive iff R (the diagonal relation) is a subset of S, that is if S includes the diagonal.

    so the relation ≤ is reflexive, but < is not (neither is >). for your relation, we can draw a partial picture like so:

    4 * _ * _
    3 _ * _ *
    2 * _ * _
    1 _ * _ *
    ...1 2 3 4 ...the *'s indicate pairs in the relation, and the _'s indicate pairs not in the relation. as you can see (visually) nothing on the diagonal is in the relation.
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