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Thread: Prove that R is an equivalence relation. Determine the distinct equivalence classes.

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    Prove that R is an equivalence relation. Determine the distinct equivalence classes.

    Course: Foundations of Higher Math

    A relation $\displaystyle R$ is defined on $\displaystyle \mathbb{Z}$ by x R y if $\displaystyle 3x-7y$ is even. Prove that $\displaystyle R$ is an equivalence relation. Determine the distinct equivalence classes.

    Reflexivity:
    Let $\displaystyle x \in \mathbb{Z}$.
    $\displaystyle 3x-7x=-4x=2(-2x)$ which is even, so x R x.

    Symmetry:
    Let $\displaystyle x,y \in \mathbb{Z}$.
    Assume that x R y ,i.e. $\displaystyle 3x-7y=2a$, $\displaystyle a\in \mathbb{Z}$.

    $\displaystyle 3x-10x-7y+10y=2a-10x+10y$
    $\displaystyle 3y-7x=2a-10x+10y$
    which is even, so y R x.

    Is this method fine?

    Also, how would I notate the equivalence classes? I believe that there will be 2 distinct classes because 3x and 7y must have same parity for the difference to be even.

    1. $\displaystyle \left\{2k|k\in\mathbb{Z}\right\}$
    2. $\displaystyle \left\{2k+1|k\in\mathbb{Z}\right\}$?
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    Re: Prove that R is an equivalence relation. Determine the distinct equivalence class

    Quote Originally Posted by MadSoulz View Post
    Course: Foundations of Higher Math

    A relation $\displaystyle R$ is defined on $\displaystyle \mathbb{Z}$ by x R y if $\displaystyle 3x-7y$ is even. Prove that $\displaystyle R$ is an equivalence relation. Determine the distinct equivalence classes.

    Reflexivity:
    Let $\displaystyle x \in \mathbb{Z}$.
    $\displaystyle 3x-7x=-4x=2(-2x)$ which is even, so x R x.

    Symmetry:
    Let $\displaystyle x,y \in \mathbb{Z}$.
    Assume that x R y ,i.e. $\displaystyle 3x-7y=2a$, $\displaystyle a\in \mathbb{Z}$.

    $\displaystyle 3x-10x-7y+10y=2a-10x+10y$
    $\displaystyle 3y-7x=2a-10x+10y$
    which is even, so y R x.

    Is this method fine?
    So far but you haven't finished. You still have to prove "transitivity": if x R y and y R z then x R z.

    Also, how would I notate the equivalence classes? I believe that there will be 2 distinct classes because 3x and 7y must have same parity for the difference to be even.

    1. $\displaystyle \left\{2k|k\in\mathbb{Z}\right\}$
    2. $\displaystyle \left\{2k+1|k\in\mathbb{Z}\right\}$?
    So you are saying that all odd numbers are equivalent and all even numbers are equivalent?
    Yes that is correct and your notation is good.
    Last edited by HallsofIvy; Nov 17th 2013 at 03:45 PM.
    Thanks from MadSoulz
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    Re: Prove that R is an equivalence relation. Determine the distinct equivalence class

    Quote Originally Posted by HallsofIvy View Post
    You still have to prove "transitivity": if x R y and y R z then x R z..
    Transitivity:
    Let $\displaystyle x,y,z \in \mathbb{Z}$
    Assume that x R y and y R z,
    i.e. $\displaystyle 3x-7y=2a$ and $\displaystyle 3y-7z=2b$. $\displaystyle a,b \in \mathbb{Z}$

    $\displaystyle 3y-7z=2b$
    $\displaystyle \Rightarrow 7y - 7z=2b +4y$
    $\displaystyle (3x-7y) + (7y-7z) = 3x-7z = 2a + 2b+4y$ which is even. So, x R z

    Hence R is an equivalence relation
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