Let X be a set containing n elements. The number of reflexive relations that can be defined on X  is

Let X=x1,x2,…xn The set X×X contains n2 elements. A reflexive relation R on X must contain xi,xi  for all 1≤i≤n,i.e.,R must contain the n elements x1,x1,x2,x2 …x0,x0 and any subset of the set containing remaining  n2–n elements. Therefore, the number of reflexive relations that can be defined on X is 2n2−n