Number of Relations on a Finite Set
Question
If \( R \) is a relation on a finite set having \( n \) elements, then the number of relations on \( A \) is
(a) \( 2^n \)
(b) \( 2^{n^2} \)
(c) \( n^2 \)
(d) \( n^n \)
Solution
Number of elements in \[ A\times A \] is \[ n^2 \]
A relation on \( A \) is any subset of \[ A\times A \]
Therefore, number of relations on \( A \) is \[ 2^{n^2} \]
Hence, the correct answer is (b).