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QuestionMathsClass 12

Show that the relation $$R$$ on the set $$Z$$ of integers, given by $$R=\left\{ \left( a,b \right) :2\ \text{divides}\ a-b \right\}$$ is an equivalence relation.

The relation $$R$$ on $$Z$$ is given by $$R=\left\{ \left( a,b \right) :2\quad \text{divides}\quad a-b \right\}$$
We observe the following properties of relation $$R$$

Reflexivity: for any $$a\in Z$$
$$a-a=0=0\times 2$$
$$\Rightarrow$$ $$2$$ divides $$a-a\Rightarrow (a,a)\in R$$
So, $$R$$ is relexive relation on $$Z$$

Symmetry: Let $$a,b\in Z$$ be such that
$$(a,b)\in R$$
$$\Rightarrow$$ $$2$$ divides $$a-b$$
$$\Rightarrow$$ $$a-b=2\lambda$$ for some $$\lambda \in Z$$
$$\Rightarrow$$ $$b-a=2(-\lambda)$$, where $$-\lambda \in Z$$
$$\Rightarrow$$ $$2$$ divides $$b-a\Rightarrow (b,a)\in R$$
Thus $$(a,b)\in R\Rightarrow (b,a)\in R$$. So, $$R$$ is a symmetric relation on $$Z$$.

Transitivity. Let $$a,b,c\in Z$$ be such that $$(a,b)\in R$$ and $$(b,c)\in R$$. Then,
$$(a,b)\in R\Rightarrow 2$$ divides $$a-b\Rightarrow a-b=2\lambda$$ for some $$\lambda \in Z$$
and $$(b,c)\in R\Rightarrow 2$$ divides $$b-c\Rightarrow b-c=2\mu$$ for some $$\mu \in Z$$
$$\therefore$$ $$a-b+b-c=2(\lambda +\mu)$$
$$\Rightarrow$$ $$a-c=2(\lambda +\mu)$$, where $$\lambda +\mu \in Z$$
$$\Rightarrow$$ $$2$$ divides $$a-c$$
$$\Rightarrow$$ $$(a,c)\in R$$
thus, $$(a,b)\in R$$ and $$(b,c)\in R\Rightarrow (a,c)\in R$$
So, $$R$$ is a transitive relation on $$Z$$
Hence, $$R$$ is an equivalence relation on $$Z$$          