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Predicate logic quesion

What methods can you use for justifications? Can you use informal reasoning, or should you use some kind of axiomatic proof?

In any case, do you have any intuition as which pairs are/aren't equivalent?

If you think a pair is NOT equivalent, then you can give a counter example.

If you think a pair IS equivalent, then you should be able to prove an implication in both directions.

But tell me your intuition first.

Cheers -- sylas

PS. Most computers should display the characters okay. They are Unicode characters 0x2200 (∀) and 0x2203 (∃). You need a font that includes them, and the capacity to manage unicode.