Question

Let $Z$ be the set of integers. Then the relation $R=\left\{\right(a,b):a,b\in Zand(a+b)iseven\}$ defined on $Z$ is

• A. Only symmetric
• B. Symmetric and transitive only
• C. An equivalence relation
• D. None of the above

Answer 1

The correct option is C

An equivalence relation

Explanation of the Correct Option:

The correct option is (C) An Equivalence Relation

Equivalence relation:

To show that a relation is Equivalence we need to show that it is reflexive, symmetric and transitive

$R$ is reflexive as $(a,a)\in R,a+a=2a$ which is even.

$R$ is symmetric as $(a,b)\in R\Rightarrow (b,a)\in R$ as $a+b=b+a=even$ (Commutative law)

$R$ is transitive as $(a,b)\in Rand(b,c)\in R$

$\Rightarrow (a+b)$ is even and $(b+c)$ is even

$\Rightarrow (a+b)+(b+c)$ is even

$\Rightarrow (a+2b+c)$ is even

$\Rightarrow (a+c)$ is even as $2b$ is even

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

$\therefore R$ is an equivalence relation on the set of integers.

Therefore , the Correct Option is (C) An Equivalence Relation