functions - To prove mapping f is injective and the other f is bijective

There's a mapping $f:X\rightarrow Y$.

1.for all $A,B\subset X$, $f(A\cap B)=f(A)\cap f(B)$, prove $f$ is injective.

2.for all $A\subset X$, $f(A^{c})=[f(A)]^{c}$, prove $f$ is bijective.

Answer

For 1 you want to show that $f(x) = f(y)$ implies $x = y$. So let $f(x) = f(y)$. Then $f(\{x\}) = f(\{y\})$ and hence $f(\{x\}) \cap f(\{y\}) = f(\{x\}) = f(\{y\}) = f(\{x\} \cap \{y\})$. Hence $x=y$.