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  4. The relation R in the set Z of integers given by R= (a, b): 2 divides a-b . The relation defined above is

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Q. The relation $R$ in the set $Z$ of integers given by $R=\{(a, b): 2$ divides $a-b\}$.
The relation defined above is

Relations and Functions - Part 2

Solution:

$R$ is reflexive, as 2 divides $(a-a)$ for all $a \in Z$. Further, if $(a, b) \in R$,then 2 divides $a-b$. Therefore, 2 divides $b-a$. Hence, $(b, a) \in R$, which shows that $R$ is symmetric. Similarly, if $(a, b) \in R$ and $(b, c) \in R$, then $a-b$ and $b-c$ are divisible by 2. Now $a-c=(a-b)+(b-c)$ is even. So, $(a-c)$ is divisible by 2 . This shows that $R$ is transitive. Thus, $R$ is an equivalence relation in $Z$.