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i was just rewriting so i could read it, i am not sure i know how to do it

well one way is trivial, since \(f(A\cap B)\subset f(A)\cap f(B)\) for any \(f\)

or does that need clarification as well? we can write it out if you like

yes we need to right it..

ok

why letter is not separated

this shows for any \(f\) you have \(f(A\cap B)\subset f(A)\cap f(B)\)

typo there, i meant "likewise there exists \(x_2\in B\) with \(f(x_2)=z\) sorry

so that is the proof one way, that "if \(f\) is injective, then \(f(A\cap B)=f(A)\cap f(B)\)

other way is easier, since a singleton is a set

A={x}
B={y}
you mean like this one..