Question

Let R and S be two relations on set A. Then

• A. $R,S$ are transitive then $R\cup S$ is also transitive
• B. $R,S$ are transitive then $R\cap S$ is also transitive
• C. $R,S$ are reflexive then $R\cap S$ is also reflexive
• D. $R,S$ are symmetric then $R\cup S$ is also symmetric
  
• E. B,C,D are correct

Answer 1

The correct option is A

$R,S$ are transitive then $R\cup S$ is also transitive

Explanation for the correct option:

We know that,
Equivalence relation means reflexive, symmetric and transitive.
Given R, and S are relations on set A
$R\subseteq A\times AandS\subseteq A\times A$
$$\Rightarrow R\cap S\subseteq A\times A$$
$$\Rightarrow R\cap S$$is also a relation on A
Reflexivity: Let a be an arbitrary element of A.
Then $$a\in A$$
$$\Rightarrow \left(a,a\right)\in R$$and$$\left(a,a\right)\in S$$.... Therefore R and S are reflexive.
$$\Rightarrow \left(a,a\right)\in R\cap S$$
Thus,$$\left(a,a\right)\in R\cap S$$for all $$a\in A$$
So$$R\cap S$$is a reflexive relation on A
$$\therefore R\cap S$$is reflexive$------\left(1\right)$
Symmetry: Let $$a,b\in A$$such that$$\left(a,b\right)\in R\cap S$$
$$\Rightarrow \left(a,b\right)\in R$$and$$\left(a,b\right)\in S$$
$$\Rightarrow \left(b,a\right)\in R$$and$$\left(b,a\right)\in S$$
( Since R and S are symmetric),
$$\Rightarrow \left(b,a\right)\in R\cap S$$
Thus,
$$\left(a,b\right)\in R\cap S$$
$$\Rightarrow \left(b,a\right)\in R\cap S$$for all $$\left(a,b\right)\in R\cap S$$.
$$R\cap S$$is symmetric on A
$$\therefore R\cap S$$is symmetric$-----\left(2\right)$
Transitivity: Let $$a,b,c\in A$$such that $$\left(a,b\right),\left(b,c\right)\in R\cap S$$
$$\Rightarrow \left(a,b\right),\left(b,c\right)\in R$$
$$\Rightarrow \left(a,c\right)\in R$$
And,
$$\Rightarrow \left(a,b\right),\left(b,c\right)\in S$$
$$\Rightarrow \left(a,c\right)\in S$$
Since R and S are transitive,
$$\Rightarrow \left(a,c\right)\in R\cap S$$
Thus,
$$\left(a,b\right)\in R\cap S$$ and $$\left(b,c\right)\in R\cap S$$

$$\Rightarrow \left(a,c\right)\in R\cap S$$
So $$R\cap S$$ is transitive on Ais transitive $---\left(3\right)$
From (1), (2) and (3), R∩S is an equivalence relation.

Hence, option(A) is the correct answer.