Q.If $n(A) = 3$ and $n(B) = 4$, find the total number of relations that can be defined from $A$ to $B$.

A relation on a set $A$ is any subset of $A \times A$, so the count of relations is $2^{|A\times A|}$.

A relation on a set $A$ is defined as a subset of the Cartesian product $A \times A$. If $A$ has $5$ elements, then $A \times A$ has $5 \times 5 = 25$ ordered pairs.

Since a relation can be any subset of this set of $25$ pairs, and a set with $n$ elements has $2^n$ subsets, the number of relations on $A$ is $2^{25}$.

A relation from $A$ to $B$ is any subset of $A\times B$; the count of subsets of a set of size $m$ is $2^m$.

Step 1. $A=\{1,2,3\}$ has $3$ elements, $B=\{a,b\}$ has $2$ elements, so $A\times B$ has $3\times2=6$ ordered pairs.

Step 2. A relation from $A$ to $B$ is any subset of $A\times B$ (including empty and full). The number of subsets of a set with $k$ elements is $2^k$.

Step 3. Here $k=6$, so the number of relations is $2^6=64$.