Originally Posted by

**elemental** I recently had problem set which asked to prove that if for any $\displaystyle A, B \subset X$, we have

$\displaystyle f(A \cap B) = f(A) \cap f(B)$ then $\displaystyle f$ is an injection. (EDITED**)

My proof started with the intention to show that if (pointwise) $\displaystyle f(A) = f(B), then A = B$. And I did that, but when I got it returned, apparently I didn't use the definition of an injection since injectivity implies that for all $\displaystyle x1, x2 \in X, f(x1) = f(x2)$ implies $\displaystyle x1 = x2$.

However, I feel that my proof was valid since A and B can have very well been singleton sets with x1 and x2, respectively. Is my approach incorrect?