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Math Help - Equivalence relation - Congruence modulo

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    Equivalence relation - Congruence modulo

    Equivalence relation - Congruence modulo-probmath-jpeg.jpg
    Last edited by aprilrocks92; November 12th 2012 at 11:30 PM.
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    Re: Equivalence relation - Congruence modulo

    What have you tried so far? You should prove by definition that the given relation is reflexive, symmetric and transitive. Can you do that?
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    Re: Equivalence relation - Congruence modulo

    Thank you. I am familiar with the properties reflexive, symmetric and transitive, but not when it comes to modulo. I have never seen it before, and simply do not know where to start.
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    MHF Contributor Siron's Avatar
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    Re: Equivalence relation - Congruence modulo

    The relation \equiv_5 is defined as \forall x,y \in \mathbb{S}_7: x\equiv_5 y \Leftrightarrow x \mod 5 = y \mod 5
    To check if the relation is reflexive you have to check \forall x \in \mathbb{S}_7: x \equiv_5 x which is true because x \mod 5 = x \mod 5.

    Can you check the symmetric and transitive property now?
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    Re: Equivalence relation - Congruence modulo

    Quote Originally Posted by aprilrocks92 View Post
    Click image for larger version. 

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    I have a different but equidistant way of describing that relation.
    Say x\mathcal{R}y if and only if x~\&~y have the same remainder when divided by 5

    Thus it should be clear that 2\mathcal{R}7.

    The three needed properties are easily checked.

    There are five equivalence classes.
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