A relation that is symmetric and transitive must also be reflexive.

Solution

Maths sir · 4 steps

Symmetry and transitivity give (a, a) only to an a with a partner; a lonely element gets nothing.

1. The tempting argument. If (a, b) ∈ R, symmetry gives (b, a) ∈ R, and then transitivity on (a, b) and (b, a) gives (a, a) ∈ R. This works only for an a that is related to at least one element.
2. The gap. Reflexive needs (a, a) ∈ R for every a in the set. An element related to nothing never gets (a, a) from this argument.
3. Counterexample. Take A = {1, 2} and R = {(1, 1)}. It is symmetric (the only pair is its own reverse) and transitive ((1,1) and (1,1) give (1,1)). But (2, 2) ∉ R, so R is not reflexive.
4. So. Symmetric and transitive do not force reflexive, so the statement is false.

The idea to keep. Reflexive is a promise about every element of the set. Symmetric and transitive only say something about elements that already appear in some pair.

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