Could someone help me prove this? This is one of the opening exercises in my Algebra book that I just can't figure out how to solve. I don't understand how to show that the equivalence classes defined by the relation would be disjoint. The opposite direction also completely baffled me on how to start. But seeing as how Wikipedia refers to this proof as the Fundamental Theorem of Equivalence Relations, and seeing how someone here or on MathOverflow commented that it was one the most important things they'd make sure an algebra student took away from their classes, I figured it's important enough for me to see it proved.
I'd really appreciate that any answers do everything with function notation, instead of using kind of slippery words like "choose" or "pick"; I'm always bad at knowing when to invoke them.
EDIT: Sorry to pull off a rush edit, but I'd also appreciate examples of what this result is useful for proving.