Question

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

• A. R ∪ S is an equivalence relation on A
• B. R ∩ S is an equivalence relation on A
• C. $R - S$ is an equivalence relation on A
• D. None of these

Answer 1

Answer: (B)

R ∩ S is an equivalence relation on A

Answer 2

Given, R and S are relations on set A.

∴ $R \cap C \subseteq A \times A$

⇒ $R \cap S$is also a relation on A.

Reflexivity : Let a be an arbitrary element of A. Then, $a \in A$⇒ $( a , a ) \in R$and $( a , a ) \in S$ , [$\bullet \bullet$R and S are reflexive]

⇒ $( a , a ) \in R \cap S$

Thus, $( a , a ) \in R \cap S$for all $a \in A$.

So, $R \cap S$is a reflexive relation on A.

Symmetry : Let $a , b \in A$such that $( a , b ) \in R \cap S$ .

Then, $( a , b ) \in R \cap S$ ⇒ $( a , b ) \in R$and $( a , b ) \in S$

⇒ $( b , a ) \in R$ and $( b , a ) \in S$ , [$\bullet \bullet$R and S are symmetric]

⇒ $( b , a ) \in R \cap S$

Thus, $( a , b ) \in R \cap S$

⇒ $( b , a ) \in R \cap S$ for all $( a , b ) \in R \cap S$ .

So, $R \cap S$is symmetric on A.

Transitivity : Let $( b , c ) \in R \cap S$

⇒ $\{ ( ( a , b ) \in R$ and $( a , b ) \in S ) \}$ and $\{ ( ( b , c ) \in R$ and $( b , c ) \in S \}$

⇒ $\{ ( a , b ) \in R , ( b , c ) \in R \}$and $\{ ( a , b ) \in S , ( b , c ) \in S \}$

⇒ $( a , c ) \in R$and $( a , c ) \in S$ $\left[ \begin{array} { l } \because R \text { and } S \text { are transitive So } \\ ( a , b ) \in R \text { and } ( b , c ) \in R \Rightarrow ( a , c ) \in R \\ ( a , b ) \in S \text { and } ( b , c ) \in S \Rightarrow ( a , c ) \in S \end{array} \right.$

⇒ $( a , c ) \in R \cap S$

Thus, $( a , b ) \in R \cap S$ and$( b , c ) \in R \cap S \Rightarrow ( a , c ) \in R \cap S$.

So, $R \cap S$is transitive on A.

Hence, R is an equivalence relation on A.