Let n(A) = m and n(B) = n. Then the total number of non-empty relations that can be defined from A to B is
Question
Let \( n(A) = m \) and \( n(B) = n \). Then the total number of non-empty relations that can be defined from \( A \) to \( B \) is ……………………
Solution
A relation from set \( A \) to set \( B \) is any subset of the Cartesian product \( A \times B \).
Number of elements in \( A \times B \):
\[ n(A \times B)=n(A)\times n(B)=mn \]
Total number of subsets of a set having \( mn \) elements is:
\[ 2^{mn} \]
Since the question asks for the number of non-empty relations, exclude the empty set.
Therefore,
\[ 2^{mn}-1 \]
Hence, the total number of non-empty relations from \( A \) to \( B \) is:
\[ \boxed{2^{mn}-1} \]