The total number of reflexive relations on a finite set having n elements is ________

The total number of reflexive relations on a finite set having n elements is 2^(n(n-1)).

Here's the step-by-step explanation:

1. A relation R on a set A is said to be reflexive if every element of A is related to itself. In other words, for every a in A, (a, a) must be in R.

2. If A has n elements, there are n(n-1) pairs of distinct elements. Each of these pairs may or may not belong to the relation R. Therefore, there are 2^(n(n-1)) ways to choose which of these pairs belong to R.

3. However, since R is reflexive, all n pairs of the form (a, a) must belong to R. Therefore, the total number of reflexive relations on A is 2^(n(n-1)).

This is because we have a choice (either to include or not) for each of the n(n-1) pairs of distinct elements, but no choice for the n pairs where both elements are the same.