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0 $begingroup$ Let $f:Ato B$ be a function.. Consider the following functions on the powersets of $A$ and $B$ : $$f[-]:mathcal P(A)tomathcal P(B), f[S] = {yin B | exists xin S, f(x)=y}$$ $$f^{-1}[-]:mathcal P(B)tomathcal P(A), f^{-1}[T] = {xin A | f(x)in T}$$ Show that for every $S,S'subseteq A, T,T' subseteq B$ : $Ssubseteq f^{-1}[f[S]]$ and $f[f^{-1}[T]]subseteq T$ My attempt: $f^{-1}[f[S]]={xin A|f(x)in{yin B|exists xin S, f(x)=y}}$ . And this for me looks pretty obvious that $S$ must be included in this set, but how do I really prove it? What's there to say about it? elementary-set-theory share | cite | improve this question ed...