The number of symmetric relations defined on the set 1, 2, 3, 4...

We know that,

Total number of relation both symmetric and reflexive $= 2\dfrac{{n}^{2} - n}{2}$, where $n$ is number of elements in given set,

Also, total number of symmetric relation $= {2}^{\left(\dfrac{{n}^{2} + n}{2}\right)}$

Then number of symmetric relation which are not reflexive will be,

$= 2\dfrac{n ( n + 1 )}{2} - 2\dfrac{n ( n - 1 )}{2}$

$= {2}^{10} - {2}^{6}$

$= 1024 - 64$

$= 960$