Class JEE Mathematics Sets, Relations, and Functions Q #1032
KNOWLEDGE BASED
APPLY
4 Marks 2023 JEE Main 2023 (Online) 1st February Evening Shift MCQ SINGLE
Let $P(S)$ denote the power set of $S=${$1, 2, 3, …, 10$}. Define the relations $R_1$ and $R_2$ on $P(S)$ as $AR_1B$ if $(A \cap B^c) \cup (B \cap A^c) = \emptyset$ and $AR_2B$ if $A \cup B^c = B \cup A^c$, $\forall A, B \in P(S)$. Then :
(A) only $R_2$ is an equivalence relation
(B) both $R_1$ and $R_2$ are not equivalence relations
(C) both $R_1$ and $R_2$ are equivalence relations
(D) only $R_1$ is an equivalence relation
Prev Next
Correct Answer: C
Explanation
$S = \{1, 2, 3, ..., 10\}$
$P(S) =$ power set of $S$
$AR_1B \implies (A \cap B^c) \cup (A^c \cap B) = \emptyset$
$R_1$ is reflexive, symmetric
For transitive:
$(A \cap B^c) \cup (A^c \cap B) = \emptyset$; $\emptyset = \emptyset \implies A = B$
$(B \cap C^c) \cup (B^c \cap C) = \emptyset \implies B = C$
$\therefore A = C \implies R_1$ is an equivalence relation.
$R_2 \equiv A \cup B^c = A^c \cup B$
$R_2 \implies$ Reflexive, symmetric
For transitive:
$A \cup B^c = A^c \cup B \implies A = B$
$B \cup C^c = B^c \cup C \implies B = C$
$\therefore A = C \implies A \cup C^c = A^c \cup C \implies R_2$ is an equivalence relation.

More from this Chapter

NUMERICAL
Let $A=\{1,2,3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is _________.
NUMERICAL
The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation, is __________.
MCQ_SINGLE
Let $A = {0, 1, 2, 3, 4, 5}$. Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $\max{x, y} \in {3, 4}$. Then among the statements (S1): The number of elements in $R$ is $18$, and (S2): The relation $R$ is symmetric but neither reflexive nor transitive
MCQ_SINGLE
Let $R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}$ be a relation on the set $A = \{3, 6, 9, 12\}$. The relation is :
MCQ_SINGLE
Let $R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$ be a relation on the set $A = \{1, 2, 3, 4\}$. The relation $R$ is:
View All Questions