ok i think i finally finished this exercise (3.4.2 in Tao’s Analysis). the “what, in general, can one say” wording is very sneaky and i initially thought that S = f^-1(f(S)) <=> f is injective. but it’s only => and not <=>. i think this is correct now. math heads am i right? 
*well <= in the order i wrote them in the tweet. but the text on the picture should be right i think?
i think maybe there’s a stronger version that would work, like if for EVERY subset S out of X, S = f^−1(f(S)), then f can be shown to be injective?
and then maybe if for EVERY subset U of Y, f(f^−1(U)) = U, then i think f should be surjective
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