Class JEE Mathematics Sets, Relations, and Functions Q #1014
KNOWLEDGE BASED
APPLY
4 Marks 2025 JEE Main 2025 (Online) 22nd January Morning Shift MCQ SINGLE
The number of non-empty equivalence relations on the set ${1, 2, 3}$ is :
(A) $7$
(B) $4$
(C) $5$
(D) $6$
Prev Next
Correct Answer: C
Explanation
An equivalence relation on a finite set is uniquely determined by its partition into equivalence classes. Counting the number of ways to partition the set ${1, 2, 3}$:

1. Three blocks: Each element in its own block. There is only one way: ${{1}, {2}, {3}}$.

2. Two blocks: We can have ${{1, 2}, {3}}$, ${{1, 3}, {2}}$, or ${{2, 3}, {1}}$. There are 3 ways.

3. One block: All elements together. There is only one way: ${{1, 2, 3}}$.

In total, there are $1 + 3 + 1 = 5$ distinct partitions, which means there are 5 equivalence relations on the set ${1, 2, 3}$.

More from this Chapter

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 $A = {1, 2, 3, …, 10}$ and $B = {\frac{m}{n} : m, n \in A, m < n$ and $gcd(m, n) = 1}$. Then $n(B)$ is equal to :
MCQ_SINGLE
Let $A = {x \in (0, \pi) - {\frac{\pi}{2}} : \log_{(2/\pi)} |\sin x| + \log_{(2/\pi)} |\cos x| = 2}$ and $B = {x \ge 0 : x(x-4) - 3|x-2| + 6 = 0}$. Then $n(A \cup B)$ is equal to :
MCQ_SINGLE
Consider the following two binary relations on the set $A = {a, b, c}$: $R_1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)}$ and $R_2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}$. Then:
MCQ_SINGLE
Let $A = {1, 2, 3, ..., 100}$ and $R$ be a relation on $A$ such that $R = {(a, b) : a = 2b + 1}$. Let $(a_1, a_2), (a_2, a_3), (a_3, a_4), ..., (a_k, a_{k+1})$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k , for which such a sequence exists, is equal to :
View All Questions