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The relation $R$ in the set $Z$ of integers given by $R=\{(a, b)$ $a-b$ is divisible by 5 \}is
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The correct answer is:
an equivalence relation
For reflexive:
$(a, a)=a-a=0$ is divisible by 5 . For symmetric:
If $(a-b)$ is divisible by 5 , then $b-a=-(a-b)$ is also divisible by 5 . Thus relation is symmetric.
For transitive If $(a-b)$ and $(b-c)$ is divisible by 5 Then $(\mathrm{a}-\mathrm{c})$ is also divisible by 5 .
Thusrelation is transitive $\mathrm{R}$ is an equivalence relation.
$(a, a)=a-a=0$ is divisible by 5 . For symmetric:
If $(a-b)$ is divisible by 5 , then $b-a=-(a-b)$ is also divisible by 5 . Thus relation is symmetric.
For transitive If $(a-b)$ and $(b-c)$ is divisible by 5 Then $(\mathrm{a}-\mathrm{c})$ is also divisible by 5 .
Thusrelation is transitive $\mathrm{R}$ is an equivalence relation.
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