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R be a relation on set of integers Z defined by R = {(x, y) | x-y is divisible by 3} Show that R is an equivalence relation and describe the equivalence classes.
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Consider any $a,b,c \in Z.$

  1. Since $a-a=0=3.0\Rightarrow(a-a)$ is divisible by 3.

    $\Rightarrow(a,a)\in R \Rightarrow$ is reflexive.

  2. Let $(a,b)\in R \Rightarrow (a-b)$ is divisible by 3.

    $\Rightarrow a-b=3q$ for some $q \in Z\Rightarrow b-a=3(-q)$

    $\Rightarrow (b-a)$ is divisible by 3 $\hspace{8cm}(\because q\in Z \Rightarrow -q \in Z \Rightarrow -q \in Z)$

    Thus, $(a,b)\in R \Rightarrow (b,a)\in R \Rightarrow R$ is symmetric.

  3. Let $(a,b)\in R$ and $(b,c)\in R$

    $\Rightarrow(a-b)$ is divisible by 3 and $(b-c)$ is divisible by 3

    $\Rightarrow a-b=3q$ and $b-c=3q'$ for some $q,q' \in Z$

    $\Rightarrow (a-b)+(b-c)=3(q+q') \Rightarrow a-c=3(q+q')$

    $\Rightarrow(a-c)$ is divisible by 3 $\hspace{8cm} (\because q.q' \in Z \Rightarrow q+q' \in Z)$

    $\Rightarrow (a,c) \in R$

    Thus, $(a,b)\in R$ and $(b,c) \in R \Rightarrow (a,c) \in R \Rightarrow R$ is transitive.

    Therefore, the relation R is reflexive, symmetric and transitive, and hence it is an equivalence relation.

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