The number of distinct equivalence relations on the set S = 1, 2, 3, 4, 5, 6 that contain exactly 14 elements is

Step-by-step solution

An equivalence relation on a set S partitions it into disjoint equivalence classes. If the sizes of these classes are x₁, x₂, , x_k , then the number of elements in the equivalence relation is given by _ i=1 ^ k x_i^2 . We are given that _ i=1 ^ k x_i = 6 (the total number of elements in S ) and _ i=1 ^ k x_i^2 = 14 . Let us check the possible integer partitions of 6 to find which one satisfies the sum of squares condition: If sizes are 4, 1, 1 , sum of squares is 16 + 1 + 1 = 18 . If sizes are 3, 3 , sum of square