Let R={(1,2),(2,3),(3,3)} be a relation defined on the set {1,2,3,4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Solution

Maths sir · 4 steps

1 links to 2 and 2 links to 3, so 1, 2, 3 form one class: build it, then subtract what R has.

1. Find the classes. An equivalence relation splits the set into classes, and every element of a class is related to every other. Since (1,2) and (2,3) are in R, the elements 1, 2 and 3 must share one class. Nothing links 4 to them, so the smallest choice keeps 4 alone.
2. Count the final relation. The class {1,2,3} needs all 3 × 3 = 9 ordered pairs. The class {4} needs only (4,4). So the smallest equivalence relation containing R has 9 + 1 = 10 pairs.

   pairs to add = 10 - 3 = 7

3. List them. The missing pairs are (1,1), (2,2), (4,4) for reflexivity, (2,1), (3,2) for symmetry, and (1,3), (3,1) for transitivity. That is 7 pairs.
4. So. The minimum number of pairs to add is 7, which is option (c).

The idea to keep. For the smallest equivalence relation containing R, group the elements R links into classes. A class of size k needs k² pairs. Add these up, then subtract the pairs R already has.

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