TITLE: How many equivalence relations on a set with 4 elements.
QUESTION [11 upvotes]: Let S be a set containing 4 elements (I choose {$a,b,c,d$}). How many possible equivalence relations are there?
So I started by making a list of the possible relations:
{$(a,a)(a,b)(a,c)(a,d)(b,a)(b,b). . .(d,d)$}
{$(a,a)(a,b)(a,c). . . . . . . . . . (d,c)$}
{$(a,a)(a,b)(a,c). . . . . . . . . . (d,b)$} etc. 
$\vdots$
{}
Then I looked at the first set of relations: $aRa, aRb,$ etc. then I tried to compare relations to see if they are equivalence relations.  This is where I got stuck.  I realized that there were way too many relations to go through, (without spending all week) and that I didn't want to have to look at each one separately.  Is there a better way to do this or am I on the right track?

REPLY [3 votes]: Ways of selecting any of the following elements out of 4 :
(i) 1 element : C(4,1)
(ii) 2 elements : C(4,2)
(iii) 3 elements : C(4,3)
(iv) 4 elements : C(4,4)
Total partitions = C(4,1) + C(4,2) + C(4,3) + C(4,4) = 15