how many reflexive relations but not equivalence, are in a set with 4 elements?

Question

I know that for reflexive relations on a set with n elements the formula is: $2^{(n^2-n)}$

So for a set with $4$ elements: $2^{(4^2-4)}$ = $2^{12}$

But I don't know how to find the relations that are reflexive but not equivalence.

Answer 1

The count of equivalence relations is the count how many ways you can partition four elements into equivalence classes.

Count how many ways can you select 4 distinct elements into each case?:

• 4 classes of one element each.
• 3 classes vis: one of two elements and two of one element
• 2 classes of two elements each
• 2 classes vis: one of three elements and one of one element
• one class of all four elements