Class JEE Mathematics Sets, Relations, and Functions Q #1032
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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
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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.

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