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Math Help - basic question about equivalence relations

  1. #1
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    basic question about equivalence relations

    I have read that: "nearer than 10" fails transitivity. In addition to that doesn't it fail symmetry (the example on integers)?
    Thanks in advance.
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  2. #2
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    Re: basic question about equivalence relations

    suppose we replace "nearer than 10" with something a bit less ambiguous (we will assume our relation is on integers, just to be specific about which set we are relating):

    a~b if (and only if) |a-b| < 10.

    it is clear that this is reflexive:

    a~a because |a-a| = |0| = 0 < 10.

    it IS symmetric because if a~b, then |a-b|= |b-a| < 10, so b~a.

    but what about transitivity: is it true that if:

    a~b and b~c, that a~c?

    no.

    suppose a = 4, b= 13, c = 17.

    then a~b because |a-b| = |4-13| = |-9| = 9 < 10.

    and b~c because |b-c| = |13-17| = |-4| = 4 < 10.

    however, we do NOT have a~c, because |a-c| = |4-17| = |-13| = 13 > 10.
    Thanks from topsquark and jojo7777777
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  3. #3
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    Re: basic question about equivalence relations

    I'm grateful to you for your help!
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