If A and B are two sets such that n(A) = 5 and n(B) = 7, then the total number of relations on A × B is
Question
If \( A \) and \( B \) are two sets such that \( n(A)=5 \) and \( n(B)=7 \), then the total number of relations on \( A \times B \) is ……………………….
Solution
Given:
\[ n(A)=5,\quad n(B)=7 \]
First find the number of elements in the Cartesian product \( A \times B \):
\[ n(A \times B)=n(A)\times n(B)=5\times7=35 \]
A relation on \( A \times B \) is any subset of:
\[ (A \times B)\times(A \times B) \]
Now,
\[ n\big((A \times B)\times(A \times B)\big)=35\times35=1225 \]
Total number of relations equals the number of subsets:
\[ 2^{1225} \]
Hence, the total number of relations on \( A \times B \) is:
\[ \boxed{2^{1225}} \]