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Math Help - Equivalence relation help

  1. #1
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    Equivalence relation help

    Let R be the relation on integers defined by aRb if and only if 2a+3b congruent to 0 (mod 5). Show that this is an equivalence relation.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by jconfer View Post
    Let R be the relation on integers defined by aRb if and only if 2a+3b congruent to 0 (mod 5). Show that this is an equivalence relation.
    Did you start by trying to show this relation has the charcteristics of an equivalence relation?
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  3. #3
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    Yeah i got the reflexive one, but i couldn't get transitive or symetry to work.

    Thats what i really need help on.
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by jconfer View Post
    Yeah i got the reflexive one, but i couldn't get transitive or symetry to work.

    Thats what i really need help on.
    ok, symmetry is a bit difficult, i'l admit when seeing this for the first time. so i will do it for you. but i will leave transitivity to you. just express the congruence as i did.


    Note that 2a + 3b \equiv 0~\mbox{mod }5 means 2a + 3b = 5k for some k \in \mathbb{Z}

    Claim: R is symmetric

    Proof: Assume aRb, that is, 2a + 3b = 5k for some integer k

    Then, 2b + 3a = 5b - 3b + 5a - 2a

    \Rightarrow 2b + 3a = 5a + 5b - \underbrace{(2a + 3b)}_{5k}

    \Rightarrow 2b + 3a = 5(a + b - k)

    since (a + b - k) \in \mathbb{Z}, we have bRa

    Thus, R is symmetric
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