The number of symmetric relations on set A=1,2,3 is

The correct answer is Option (4) → 64

A relation on A is a subset of \( A\times A \), which has 9 ordered pairs. It is symmetric when (a, b) ∈ R always comes with (b, a) ∈ R.

So each of the 3 diagonal pairs (1, 1), (2, 2), (3, 3) can be taken or left freely, and each of the 3 couples {(1, 2), (2, 1)}, {(1, 3), (3, 1)}, {(2, 3), (3, 2)} must be taken or left together:

    \( 2^{3+3}=2^{6} \)

    \( =64 \)

Tip: A set with n elements has \( 2^{n(n+1)/2} \) symmetric relations; for n = 3 this is \( 2^{6}=64 \).