Problem Statement:

Suppose R is a relation on A and let S be the transitive closure of R. Prove that Dom(S) = Dom(R) and Ran(S) = Ran(R).

*Dom() signifies the domain of the specified relation. Ran() signifies the range of the specified relation.*

My attempt thus far:

Let M = {T ⊆ A x A | R ⊆ T and T is transitive}. I have already shown (in a different exercise) that ∩M is the transitive closure of R (so S = ∩M).

Let x ∈ Dom(S). Then there is some y ∈ A such that (x,y) ∈ S = ∩M. This means (∀K ∈ M)((x,y) ∈ K). We must show x ∈ Dom(R), which means there is some z ∈ A such that (x,z) ∈ R... (and this is where i can't seem to find how to proceed)

As you can see, I am trying to prove the two conjuncts separately. I am proving the equalities by proving that each side is a subset of the other. I can prove the opposite direction- Dom(R) ⊆ Dom(S)- easily, but this one doesn't seem as easy. I'm sure it is analogous for the range equalities as well.

Any insight is appreciated, thanks!