Let R and S be two equivalence relations on a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) An equivalence relation must be reflexive,symmetric,and transitive. $1$. Reflexivity: Since $R$ and $S$ are equivalence relations,$(a, a) \in R$ a

Vedclass · Accepted Answer

$R \cap S$ is an equivalence relation

Vedclass · Answer

(B) An equivalence relation must be reflexive,symmetric,and transitive. $1$. Reflexivity: Since $R$ and $S$ are equivalence relations,$(a, a) \in R$ and $(a, a) \in S$ for all $a \in A$. Thus,$(a, a) \in R \cap S$. $2$. Symmetry: If $(a, b) \in R \cap S$,then $(a, b) \in R$ and $(a, b) \in S$. Since $R$ and $S$ are symmetric,$(b, a) \in R$ and $(b, a) \in S$,so $(b, a) \in R \cap S$. $3$. Transitivity: If $(a, b) \in R \cap S$ and $(b, c) \in R \cap S$,then $(a, b), (b, c) \in R$ and $(a, b), (b, c) \in S$. Since $R$ and $S$ are transitive,$(a, c) \in R$ and $(a, c) \in S$,so $(a, c) \in R \cap S$. Therefore,$R \cap S$ is an equivalence relation.

Let $R$ and $S$ be two equivalence relations on a non-void set $A$. Then

Similar Questions

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time given by $R = \{(x, y): x \text{ and } y \text{ work at the same place}\}$.

Let $A = \{1, 2, 3\}$. What is the total number of relations defined on $A$?

Let $A = \{1, 2, 3, 4\}$ and let $R = \{(2, 2), (3, 3), (4, 4), (1, 2)\}$ be a relation on $A$. Then $R$ is

The void relation on a set $A$ is

Let ${R_1}$ be a relation defined by ${R_1} = \{ (a, b) | a \ge b, a, b \in R \}$. Then ${R_1}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module