If the numbers of different reflexive relations on a set A is equal to the number of different symmetric relations on set A, then the numbers of elements in A is _____.

Let there be n elements in set A. Then,

Number of different reflexive relations on A = $2^{n^2-n}$

Number of different symmetric relations on A = $A=2^{\frac{n^2+n}{2}}$

It is given that

$2^{n^2-n}=2^{\frac{n^2+n}{2}}$

$⇒n^2-n=\frac{n^2+n}{2}⇒n^2-3n=0⇒n(n-3)=0⇒n=3$