Hello again, I'm back with one more proof question, this time concerning equivalence relations. I have an okay time conceptualizing what is supposed to be going on in the proof, but actually laying everything out in a manner that makes sense / is correct is tough for me.

I feel fairly confident with showing that is reflexive, but in my other steps, equating things to like that just made it seem to easy, as if I perhaps missed something vital.

Anyway, here's what I've done so far for this proof:

Theorem: Let be a function. Define a relation on via iff . Then is an equivalence relation.

Proof: Let be a function. Define a relation on via iff .

Reflexive: Let . We see that . So .

Symmetric: Let where and for some . We see that and similarly . So .

Transitive: Let where , , and for some . We see that . We also see that . Transitively, we see that . So .

Therfore, by definition, is an equivalence relation.