Confusingly enough, "transitive
closure of a
set" means something completely different to "transitive
closure of a
relation".
Recall that a set A is said to be transitive if whenever x is in A and y is in x then y is in A. To put that another way, every member is also a subset.
If X is any set, the transitive closure of X is the smallest set Y such that Y is transitive and contains X. "Smallest" in this context means that whenever Z is transitive and contains X, then Z contains Y.
It can be shown that every set has a transitive closure; this is an application of the recursion theorem.