Number of Relations from Set A to Set B
Question
If \( R \) is a relation from a finite set \( A \) having \( m \) elements to a finite set \( B \) having \( n \) elements, then the number of relations from \( A \) to \( B \) is
(a) \( 2^{mn} \)
(b) \( 2^{mn}-1 \)
(c) \( 2mn \)
(d) \( mn \)
Solution
Number of elements in \[ A\times B \] is \[ mn \]
A relation from \( A \) to \( B \) is any subset of \[ A\times B \]
Number of subsets of a set having \( mn \) elements is \[ 2^{mn} \]
Hence, the correct answer is (a).