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\(\displaystyle (A\times B)\cap (C\times D)=(A\cap C)\times (B\cap D)\)

If \(\displaystyle g\circ f\) is injective, then \(\displaystyle f\) is injective (but not necessarily \(\displaystyle g\)). If \(\displaystyle g\circ f\) is surjective, then \(\displaystyle g\) is surjective (but not necessarily \(\displaystyle f\)).

Given \(\displaystyle f:A\rightarrow B\), \(\displaystyle A_0,A_1\subset A\) and \(\displaystyle B_0,B_1\subset B\):

\(\displaystyle f^{-1}(B_0\cap B_1)=f^{-1}(B_0)\cap f^{-1}(B_1)\) and

\(\displaystyle f(A_0\cap A_1)\subset f(A_0)\cap f(A_1)\) - equality holds if \(\displaystyle f\) is injective.

The thing is, proving such things doesn't seem to help me remember them since the proof is rather mechanical and symbolic. Does anyone have any tips for learning them better, or any book suggestions that might give me some intuition for it? Thanks!