Question

Let $n\left(A\right)=n$. Then the number of all relations on $A$ is

• A. $2^n$
• B. $2^{\left(n\right)!}$
• C. $2^{n^2}$
• D. none

Answer 1

The correct option is C $2^{n^2}$

For any set $A$ such that $n\left(A\right)=n$
then number of all relations on $A$ is $2^{n^2}$

As the total number of Relations that can be defined from a set $A$ to $B$ is the number of possible subsets of $A\times B$. If $n\left(A\right)=p$ and $n\left(B\right)=q$ then $n(A\times B)=pq$ and the number of subsets of $A\times B$ = $2^{pq}$.