Let the no. of elements in a set be denoted by |S|.
Let |S| contains n elements , then |SxS| = n2
SxS = { (1,1),(1,2),(1,3)........................(1,n)
(2,1),(2,2)................................(2,n)
.......................................................
(n,1),(n,2),..............................(n,n)}
In the cross product of the set, there are n diagonal elements i.e. (1,1),(2,2)......upto (n,n). These elements can either be present in a symmetric relation or absent . Hence for n such elements, there are 2n combinations.
Among the remaining elements , i.e. n2-n, pair of elements will form a symmetric set for eg (1,2),(2,1) ; (1,3),(3,1) and so on . Hence (n2-n)/2 pairs will contribute to the formation of symmetric sets .
Hence total number of symmetric sets possible on a set with n elements would be 2n * 2(n^2-n)/2
Hence for 4 elements as given in the question, there would be 24 * 2(4^2-4)/2
= 24 * 26 = 210 symmetric sets of B