Prove that for any $A, B \subseteq X$ we have $f(A \cap B) = f(A) \cap f(B)$, then $f$ is an injection.
I get stuck at the step where $f(w) = y = f(z)$, since I am trying to prove it is injective, I can't just say they're equal right?
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1This is neither about multisets nor injective modules, I edited the tags accordingly – Hagen von Eitzen Feb 18 '14 at 16:18
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If $f(w)=f(z)=y$, then you know that $f(\{w\}\cap \{z\})=f(\{w\})\cap f(\{z\})=\{y\}$. Since $f(\emptyset)=\emptyset$, we conclude $\{w\}\cap\{z\}\ne\emptyset$, i.e. $w=z$.
Hagen von Eitzen
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