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JEE Mathematics 2025 Question with Solution

The number of non-empty equivalence relations on the set {1,2,3}\{1,2,3\} is :

  • A

    66

  • B

    77

  • C

    55

  • D

    44

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: We need the number of non-empty equivalence relations on the set {1,2,3}\{1,2,3\}.

Find: The total number of such equivalence relations.

An equivalence relation on a set is in one-to-one correspondence with a partition of that set. So we count all partitions of {1,2,3}\{1,2,3\}.

The possible partitions are:

  1. {{1,2,3}}\{\{1,2,3\}\}
  2. {{1},{2,3}}\{\{1\},\{2,3\}\}
  3. {{2},{1,3}}\{\{2\},\{1,3\}\}
  4. {{3},{1,2}}\{\{3\},\{1,2\}\}
  5. {{1},{2},{3}}\{\{1\},\{2\},\{3\}\}

Thus, the total number of partitions is 55. Hence, the number of non-empty equivalence relations on the set {1,2,3}\{1,2,3\} is 55.

Therefore, the correct option is C.

Using Bell Number

Given: We need the number of equivalence relations on a 33-element set.

Find: The required count quickly.

Equivalence relations on a set are counted by the number of partitions of that set. The number of partitions of an nn-element set is the Bell number BnB_n.

For a set with 33 elements,

B3=5B_3 = 5

Therefore, the number of equivalence relations on {1,2,3}\{1,2,3\} is 55, so the correct option is C.

Common mistakes

  • Counting only the partitions with two subsets is incorrect because equivalence relations also come from the single-block partition and the three-singleton partition. Count all partitions of the set.

  • Treating equivalence relations as arbitrary subsets of {1,2,3}×{1,2,3}\{1,2,3\} \times \{1,2,3\} is wrong because an equivalence relation must satisfy reflexivity, symmetry, and transitivity. Use the partition correspondence instead.

  • Forgetting that each partition gives exactly one equivalence relation leads to overcounting or undercounting. First list distinct partitions, then map each partition to one relation.

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