Let f : A -> B be a function and K subset of B. Prove that f(f^-1(K)) = K intersection Image(f).
How do we prove equations like these without using venn diagrams?�^^^6sugwerwerfwewe(K)) = K \ Im(f).
You prove equality $\displaystyle \scriptstyle f(f^{-1}(K))=K\cap \mathrm{Im}(f)$ of sets in two steps: first, you prove $\displaystyle \scriptstyle f(f^{-1}(K))\subseteq K\cap \mathrm{Im}(f)$, then you prove $\displaystyle \scriptstyle f(f^{-1}(K))\supseteq K\cap \mathrm{Im}(f)$.
That approach should work reasonably well for your problem, provided you know how $\displaystyle \scriptstyle f$ and $\displaystyle \scriptstyle f^{-1}$, applied to sets, are defined. $\displaystyle \scriptstyle \mathrm{Im}(f)$ is just $\displaystyle \scriptstyle f(A)$, of course.