The number of symmetric relations on the set A = a, b, c, d is

The correct answer is Option (4) → \(2^{10}\)

In a symmetric relation, a pair (x, y) with \(x \ne y\) is in R exactly when (y, x) is. So the choices are made as follows.

Each of the 4 diagonal pairs (a, a), (b, b), (c, c), (d, d) can be put in or left out on its own: 4 free choices.

The pairs with \(x \ne y\) go in as unordered pairs {x, y}, each either in (both orders) or out. Their number is

    \( \displaystyle \frac{4 \times 3}{2} = 6 \)

So there are \(4 + 6 = 10\) independent yes/no choices:

    number of symmetric relations \(= 2^{10}\)