Question

If $\text{A}$ and $\text{B}$ are two equivalence relations defined on the set $\text{C}$, then

• A. $\text{A}$ intersection $\text{B}$ is an equivalence relation
• B. $\text{A}$ intersection $\text{B}$ is not an equivalence relation
• C. $\text{A U B}$ is an equivalence relation
• D. $\text{A U B}$ is not an equivalence relation

Answer 1

The correct option is A

$\text{A}$ intersection $\text{B}$ is an equivalence relation

Explanation for the correct option :

Step $1$. Equivalence relation

A relation $\text{A}$ on a set $\text{C}$ is called an equivalence relation if it satisfies following three properties:

1. Relation $\text{A}$ is Reflexive, i.e., $\left(a,a\right)\in \text{A}$
2. Relation $\text{A}$ is Symmetric, i.e., if $\left(a,b\right)\in \text{A}$ $\Rightarrow$ $\left(b,a\right)\in \text{A}$
3. Relation $\text{A}$ is transitive, i.e., if $\left(a,b\right)\in \text{A}$ and $\left(b,c\right)\in \text{A}$ $\Rightarrow$ $\left(a,c\right)\in \text{A}$

Step $2$. Example

Example: $\text{A}$ $=$ $\left\{1,2,3\right\}$
${\text{R}}_1$ $=$ $\left\{(1,1),(2,2),(3,3),(1,2),(2,1)\right\}$
${\text{R}}_2$ $=$ $\left\{(1,1),(2,2),(3,3),(2,3),(3,2)\right\}$
$\text{R1∩ R2}$ $=$ $\left\{(1,1),(2,2),(3,3)\right\}$

Therefore, if ${\text{R}}_1$ and ${\text{R}}_2$ are equivalence relation then $\text{R1∩ R2}$ is also an equivalence relation.

and if ${\text{R}}_1$ and ${\text{R}}_2$ are equivalence relation then $\text{R1∪ R2}$ may or may not be an equivalence relation.

Hence, option $"\text{A"}$ is correct.