Problems related to Sets and Mappings

Question

I have a problem which is related to Sets and Mappings. For details:

Given a mapping: $f: X \rightarrow Y$ and $A \subset X$. The following conclusion is true or not? Why? $$ f(X) \backslash f(A) \subset f(X\backslash A) $$ I thought that this problem could be solved by using element principle, which means that using $x$ belongs to one side and prove the other but I failed. I really need your help to solve this problem.

Answer 1

Hint: If $y\in f(X)\setminus f(A)$, then $y=f(x)$ for some $x\in X$. Can it happen that $x\in A$?

No: if $x\in A$, then $y=f(x)$ would be in $f(A)$, contradicting the hypothesis $y\notin f(A)$.
So the answer on the original question is affirmative.