TEZPUR 2025 — Mathematics PYQ

An equivalence relation on a set is closely tied to the concept of partitions of that set. The number of equivalence relations on a set with $n$ elements is given by the $n$-th Bell number, denoted as $B_n$.

For a set $A$ with $n=3$ elements:

1. Understand the correspondence: Every partition of a set corresponds to exactly one equivalence relation.

2. Find the partitions of $A=\{1,2,3\}$:

   • Partition into 1 subset: $\{\{1,2,3\}\}$ — (1 way)

   • Partition into 2 subsets: $\{\{1,2\},\{3\}\},\{\{1,3\},\{2\}\},\{\{2,3\},\{1\}\}$ — (3 ways)

   • Partition into 3 subsets: $\{\{1\},\{2\},\{3\}\}$ — (1 way)

3. Summing the possibilities:

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

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

Alternatively, using the Bell number recurrence formula:

$$B_{n+1}=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)B_k$$

For $n=3$:

$$B_3=\left(\genfrac{}{}{0ex}{}{2}{0}\right)B_0+\left(\genfrac{}{}{0ex}{}{2}{1}\right)B_1+\left(\genfrac{}{}{0ex}{}{2}{2}\right)B_2$$

$$B_3=(1\times 1)+(2\times 1)+(1\times 2)=1+2+2=5$$

Conclusion:

The maximum number of equivalence relations on the set $A=\{1,2,3\}$ is $5$. The correct option is (d).