How many equivalence relations can be defined on the set $S = {1, 2, 3, 4, 5, 6}$ such that the corresponding partition has exactly two equivalence classes, one of size $2$ and the other of size $4$ ?

medium 2 marks Class 12 · Mathematics · Relations and Functions

Answer

The correct answer is 15 — Each choice yields a distinct partition, and hence a distinct equivalence relation.

Step-by-step solution

An equivalence relation corresponds uniquely to a partition of the set.
We need to count the number of ways to choose a $2$ ‑element subset; the remaining $4$ elements automatically form the second class.
The number of $2$ ‑element subsets of a $6$ ‑element set is the binomial coefficient $(2 6)$ .
Compute $(2 6) = \frac{6 \times 5}{2} = 15$ .
Each choice yields a distinct partition, and hence a distinct equivalence relation.

Common mistake: Multiplying by

2

for the order of the two classes, forgetting that the classes are unordered.

Concept tested: Counting equivalence relations via partitions