# Writing a proof for $f(W) \setminus f(X) \subseteq f(W\setminus X)$

I am trying to write a proof to prove/disprove the following question:

Will it always be true that $f(W\setminus X) = f(W)\setminus f(X)$?

I know to prove this you need to show both ways since you have = sign. I have found a counterexample for $f(W\setminus X) \subseteq f(W)\setminus f(X)$. However, I know $f(W)\setminus f(X)\subseteq f(W\setminus X)$ is true, because I've been told it's true, but I don't understand how it's true and thus don't know how to write the proof for it. Could someone please explain why it's true/show how the proof would be written?

• Your Question involves two propositions, one about equality of two sets and one about a subset relationship between the same two sets. Please clarify what you are asking versus what you know is "a counterexample". – hardmath May 18 '16 at 23:12

Let $y \in f(W) \backslash f(X)$, then $\exists w \in W, f(w)=y, w \notin X$. But this just implies that we have found $w \in W \backslash X$, such that $y=f(w).$ Hence $f(W) \backslash f(X) \subset f(W)\backslash f(X).$