No. of Equivalence Relations on a set of $n$ elements is given by the $n^{th}$ BELL number $B_n$.
$B_{n}$ is also equal to the number of different ways to partition a set that has exactly $n$ elements, or equivalently, the number of equivalence relations on it.
Ref: https://en.wikipedia.org/wiki/Bell_number
$n^{th}$ Bell number can be found easily from the Bell triangle as follows:
Here, $E_{(i,j)} = E_{(i-1,j-1)}+E_{(i,j-1)}; i,j > 1,$
$\qquad E_{(1,1)} = 1, E_{(i,1)} = E_{(i-1,i-1)}$
$\begin{array}{ccc}1&&&&-\text{ No. of partitions for 1 element set}\\1 &2&&&-\text{ No. of partitions for 2 element set}\\2&3&5&&-\text{ No. of partitions for 3 element set}\\5&7&10&15&-\text{ No. of partitions for 4 element set}\end{array}.$
So, answer is (A) 15.