Question

If the numbers of elements of two finite sets \( A \) and \( B \) are \( m \) and \( n \) respectively, then the total number of relations from \( A \) to \( B \) will be

• A. \( 2^{m+n} \)
• B. \( 2^{mn} \)
• C. \( m \times n \)
• D. \( m + n \)

Hint:

To find the total number of relations between two sets, calculate the total number of elements in the Cartesian product \( A \times B \), which is \( m \times n \), and then raise 2 to the power of \( m \times n \) to get the total number of subsets (relations).

Answer 1

The Correct Option is B

Step 1: Understanding Relations Between Two Sets
A relation between two sets \( A \) and \( B \) is a subset of the Cartesian product \( A \times B \), which consists of all ordered pairs \( (a, b) \), where \( a \in A \) and \( b \in B \). The total number of elements in \( A \times B \) is \( m \times n \), where \( m \) is the number of elements in set \( A \) and \( n \) is the number of elements in set \( B \).

Step 2: Finding the Total Number of Relations
A relation is a subset of \( A \times B \). The total number of subsets of a set with \( k \) elements is \( 2^k \), so the total number of relations is the number of subsets of \( A \times B \), which is \( 2^{m \times n} \).

Step 3: Conclusion
Thus, the total number of relations from \( A \) to \( B \) is \( 2^{mn} \). The correct answer is \( 2^{mn} \), which corresponds to option (B).