Question

How many relations are there on a set with \(n\) elements?

Short answer

There are \({2^{{n^2}}}\) relations on a set \(S\) with \(n\) elements.

Step-by-step solution

Given data

A set with \(n\) elements .

Formula used number of relation on a set

If a set\(A\)has\(n\)elements then number of relations on\(A\)is given by\({2^{{n^2}}}\).

Evaluate relations on given set

Given set is \(S = \left\{ {{a_1},{a_2},{a_3}, \ldots ,{a_n}} \right\}\).

Thus, there are \({2^{{n^2}}}\) relations on a set \(S\) with \(n\) elements.