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
The number of elements in the set $\left\{n \in \mathbb{N}: 10 \leq n \leq 100\right.$ and $3^{n}-3$ is a multiple of 7$\}$ is ___________.
NUMERICAL
Let A = {n $ \in $ N: n is a 3-digit number} B = {9k + 2: k $ \in $ N} and C = {9k + $l$: k $ \in $ N} for some $l ( 0 < l < 9)$ If the sum of all the elements of the set A $ \cap $ (B $ \cup $ C) is 274 $ \times $ 400, then $l$ is equal to ________.
NUMERICAL
Let $S=\{4,6,9\}$ and $T=\{9,10,11, \ldots, 1000\}$. If $A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\right.$ $\epsilon S\}$, then the sum of all the elements in the set $T-A$ is equal to __________.
NUMERICAL
Let $\mathrm{A}=\{1,2,3,4, \ldots ., 10\}$ and $\mathrm{B}=\{0,1,2,3,4\}$. The number of elements in the relation $R=\left\{(a, b) \in A \times A: 2(a-b)^{2}+3(a-b) \in B\right\}$ is ___________.
NUMERICAL
Let A = {n $\in$ N : H.C.F. (n, 45) = 1} and Let B = {2k : k $\in$ {1, 2, ......., 100}}. Then the sum of all the elements of A $\cap$ B is ____________.
View All Questions