Let * be a binary operation on a nonempty set X and let Y be a nonempty subset of X and be a family of nonempty subsets of X. Prove that:
For the opposite direction, I have:
Then there exist an element such that
Thus complete the proof. I hope this is right.
There is a second part of the problem:
Prove that at least one of and holds, but they don't have to equal.
Now, so that means I would have two cases. But what gives me trouble is that what two different cases would give me those two different result? Would it be that and ?