Q.Number of relations on a set with 5 elements is

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.

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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.

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