Number of Non-Empty Relations from A to B
Question
Let \[ n(A)=m \quad \text{and} \quad n(B)=n \] Then the total number of non-empty relations that can be defined from \( A \) to \( B \) is
(a) \( m^n \)
(b) \( n^m-1 \)
(c) \( mn-1 \)
(d) \( 2^{mn}-1 \)
Solution
Number of elements in \[ A\times B \] is \[ mn \]
Total number of relations from \( A \) to \( B \) is \[ 2^{mn} \]
Excluding the empty relation, \[ 2^{mn}-1 \]
Hence, the correct answer is (d).