Let $R=\{(a, b): a, b \in Z$ and $(a+b)$ is even $\}$. Show that $R$ is an equivalence relation on $Z$.

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Let $R=\{(a, b): a, b \in Z$ and $(a+b)$ is even $\}$ . Show that $R$ is an equivalence relation on $Z$ .

Let $R=\{(a, b): a, b \in Z$ and $(a+b)$ is even $\}$ . Show that $R$ is an equivalence relation on $Z$ .

Solution:

$\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathrm{Z}$ and $(\mathrm{a}+\mathrm{b})$ is even $\}$ (As given)
If $R$ is Reflexive, Symmetric and Transitive, then $R$ is an equivalence relation.

Reflexivity:
Suppose a be an arbitrary element of $Z$
$\mathrm{a}+\mathrm{a}=2 \mathrm{a}$
As $2 \mathrm{a}$ is even
$\Rightarrow(\mathrm{a}, \mathrm{a}) \in \mathrm{R}$
Therefore, $\mathrm{R}$ is reflexive.

Symmetric:
Suppose a and $\mathrm{b} \in \mathrm{Z}$ , such that $(\mathrm{a}, \mathrm{b}) \in \mathrm{R}$
$\Rightarrow \mathrm{a}+\mathrm{b}=$ even.
$\begin{array}{l} \Rightarrow \mathrm{b}+\mathrm{a}=\text { even } \\ \Rightarrow(\mathrm{b}, \mathrm{a}) \in \mathrm{R} \end{array}$
Therefore, $\mathrm{R}$ is symmetric.

Transitivity:
Suppose $\mathrm{a}, \mathrm{b}$ and $c \in \mathrm{Z}$ , such that $(\mathrm{a}, \mathrm{b}) \in \mathrm{R}$ and $(\mathrm{b}, \mathrm{c}) \in \mathrm{R}$ $\Rightarrow a+b=2 k$ (which is an even)
and $b+c=21$ (which is an even)
On adding both the above equations, we obtain
$\begin{array}{l} \Rightarrow \mathrm{a}+\mathrm{c}+2 \mathrm{~b}=2(\mathrm{k}+1) \\ \Rightarrow \mathrm{a}+\mathrm{c}=2(\mathrm{k}+1)-2 \mathrm{~b} \end{array}$
$\Rightarrow \mathrm{a}+\mathrm{c}$ is an even number
$\Rightarrow(a, c) \in R$
Therefore, $R$ is transitive.
As a result, $R$ is an equivalence relation.