The number of equivalence relations on a set with 3 elements is x.

Solution

Maths sir · 3 steps

Every equivalence relation splits the set into classes, so count the ways to split {a, b, c} into groups.

1. Relations are partitions. An equivalence relation groups elements into disjoint classes, and each grouping gives exactly one equivalence relation. So counting equivalence relations is the same as counting partitions of {a, b, c}.

   Shape of partition	Partitions	Count
   Three single classes	{a}, {b}, {c}	1
   One pair and one single	{a, b}, {c}; {a, c}, {b}; {b, c}, {a}	3
   One class of all three	{a, b, c}	1

2. Two extremes. The first row is the identity relation {(a,a), (b,b), (c,c)}. The last row is the universal relation A × A. The middle row gives three different relations, because the element left alone can be a, b or c.
3. So. The total is 1 + 3 + 1 = 5, so x = 5.

The common slip: 3. Counting only the three shapes of partition, and not the three ways to choose the pair, gives 3, not 5.

The idea to keep. Equivalence relations on a set match its partitions one to one. For small sets, list the partitions shape by shape and count how many ways each shape can be filled.

Answer from the NTA final answer key. Solution by PrepWiser.