The minimum number of elements that must be added to the relation R={(a, b),(b, c)} on the set {a, b, c} so that it becomes symmetric and transitive is :

Solution

Maths sir · 4 steps

Add each missing pair one rule at a time, and keep going until no rule asks for anything new.

1. Make it symmetric. Every (x, y) needs its reverse (y, x). So add (b, a) and (c, b). The relation now has 4 pairs.
2. Chain through b. Transitivity says (x, y) and (y, z) force (x, z). From (a, b), (b, a) we need (a, a). From (b, a), (a, b) we need (b, b). From (c, b), (b, c) we need (c, c). From (a, b), (b, c) we need (a, c).
3. Symmetry again. The new pair (a, c) needs its reverse (c, a). Now every one of the 3 × 3 = 9 ordered pairs is present, so nothing more can be required.

   added = {(b,a), (c,b), (a,a), (b,b), (c,c), (a,c), (c,a)}

4. So. R becomes the full relation A × A with 9 pairs. It started with 2, so the minimum number to add is 9 - 2 = 7, option (a).

The idea to keep. Symmetric plus transitive pulls in (x, x) for every element that is related to anything, because (x, y) and (y, x) chain back to (x, x). After every addition, check both rules again.

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