JEE Main 2025 — 22 January, Morning Shift — Question 1

Fundamental Concept There is a one-to-one correspondence between the number of equivalence relations on a set and the number of partitions of that set. The total number of partitions of a set with $n$ elements is given by the Bell Number, $B_n$.

Case-by-Case Partition. For the set $S = \{1, 2, 3\}$, where $n = 3$, we list all possible partitions:

enumerate Single Subset (All elements together):

$$\{ \{1, 2, 3\} \}$$

Number of ways: $\mathbf{1}$

Two Subsets (One pair and one singleton):

$$\{ \{1, 2\}, \{3\} \}, \quad \{ \{1, 3\}, \{2\} \}, \quad \{ \{2, 3\}, \{1\} \}$$

Number of ways: $\mathbf{3}$

Three Subsets (All singletons):

$$\{ \{1\}, \{2\}, \{3\} \}$$

Number of ways: $\mathbf{1}$

Total Count The total number of equivalence relations is the sum of these partitions:

$$B_3 = 1 + 3 + 1 = 5$$

Note on “Non-empty” An equivalence relation on a set $S \neq \emptyset$ must be reflexive. This implies that for every $a \in S$, the pair $(a, a)$ must belong to the relation. Thus, the identity relation $I = \{(1,1), (2,2), (3,3)\}$ is the smallest possible equivalence relation. Consequently, no equivalence relation on this set can be empty. The number of equivalence relations on the set $\{1, 2, 3\}$ is $5.$