KCET 2025 Math Paper

To determine the number of equivalence relations on the set A={a,b,c} that include the pair (b,c), we must ensure the relations satisfy the reflexivity, symmetry, and transitivity properties of equivalence relations.
Given:
Reflexivity requires that every element is related to itself: (a,a),(b,b), and (c,c) must be included.
Symmetry necessitates that if (b,c) is included, then (c,b) must also be included.
Transitivity requires that for the inclusion of (b,c), the set must also consider the relationships between (a,b),(b,a),(a,c), and (c,a).
There are two potential equivalence relations:
Minimal relation involving (b,c) :
{(a,a),(b,b),(c,c),(b,c),(c,b)}
Full equivalence connecting all elements:
Thus, there are 2 equivalence relations on A containing (b,c).