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Equivalence Class

Let R1 and R2...

Question

Let $R_{1}$ and $R_{2}$ be two equivalence relations on a set. Consider the following assertions:
(i) $R_{1} \cup R_{2}$ is an equivalence relation
(ii) $R_{1} \cap R_{2}$ is an equivalence relation
Which of the following is correct ?

A

Both assertions are true

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B

Assertion (i) is true but assertion (ii) is not true

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C

Assertion (ii) is true but assertion (i) is not true

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D

Neither (i) nor (ii) is true

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Solution

The correct option is C Assertion (ii) is true but assertion (i) is not true
A relation is said to be equivalence relation if relation is
(i) Reflexive
(ii) Symmetric
(iii) Transitive
Reflexive and symmetric properties are both closed under $\cup$ and $\cap$ .
Transitive property is closed under $\cap$ but not $\cup$ .
So equivalence relations are closed under $\cap$ but not $\cup$
Therefore $R_{1} \cap R_{2}$ is an equivalence relation but $R_{1} \cup R_{2}$ is not necessarily an equivalence relation.

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