Generating equivalence relations

Question

Let $\sim$ be the equivalence relation generated by a relation $x\sim y$ on a non empty set X. Show that the equivalence relation always exists.

Note that I think I am supposed to show that the smallest equivalence relation which contains the set $\{$ $(x,y)$ : $x,y\in$ $\sim$ $\}$ exists.

How would i show that ? May I have hints?

Answer 1

Here is the general structure of what I presume is the standard proof:

1. Show that there is at least one equivalence relation that contains $\sim$ (most easily done by constructing one such relation).
2. Note that because of 1., the set of all equivalence relations that contain $\sim$ is non-empty.
3. Show that the intersection of all these equivalence relations is an equivalence relation.
4. Show that this intersection is indeed the smallest equivalence relation that contains $\sim$.