Question

Assertion :The relation $R=A\times A$ ( where $A$ is non empty set) is an equivalence relation. Reason: Relation $R=A\times A$ satisfies the property reflexivety , symmetricity and transitivity.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $x\in A\therefore (x,x)\in A\times A$

$xRx\Rightarrow$ is reflexive.

Let $(x,y)\in R,x,y\in A$

$\Rightarrow (y,x)\in R\Rightarrow R$ is symmetric.

Let $(x,y),(y,z)\in R\therefore x,y,z\in A$

$\therefore (x,z)\in R\Rightarrow R$ is transitive as
$R$ is reflexive, symmetric and transitive

$\Rightarrow R$ is an equivalence relation.