Is the family of equivalence relations on a countable infinite set uncountable? 
Is the family of equivalence relations on a countable infinite set always an uncountable infinite set?

I can't seem to prove it even though intuitively I think it's true.
 A: HINT: Recall that equivalence relations are determined by their equivalence classes. Show that there are uncountably many partitions of a countably infinite set into two infinite parts. 
A: You know the number of subsets of the countable set $S$ is uncountable. If $A$ is a subset of the countable set $S$, there is a unique equivalence relation whose cells are $A$ and the complement of $A$. This gives uncountably many equivalence relations on $S$ which are only a small portion of all the equivalence relations. 
Technicality: The partitions associated to a set $A$ and its complement $A'$ are the same, so the map suggested above from subsets of $S$ to equivalence relations is actually a "two-to-one" map, so some argument is required to show that this still produces an uncountable number of distinct equivalence relations.
The "technicality" mentioned here can be easily handled. Select one specific element $a \in S$ and let $A$ be any of the (uncountably many) subsets of $S$ which happen to contain $a$. Then put $B=S \setminus A$ and associate to $A$ the unique equivalence relation with cells $A,B.$ This gives a one-to-one correspondence between subsets $A$ of $S$ for which $a \in A$, and a set of equivalence relations on $S$. Then since there are uncountably many $A$, there are also uncountably many equivalence relations.
