Difference of Function of Sets Equals Function of Difference of Sets Implies One to One

Question

Given a function $f:X\rightarrow Y$ where $A,B\subset X$, prove that if $f(A)\backslash f(B)=f(A\backslash B)$, then $f$ must be one-to-one (injective).

I'm really struggling with this, because I can't use typical properties of functions considering that $y\in f(A)\cap f(B)^C$. This has me really confused, and I'm not sure how to work with it.

Answer 1

Suppose $f(a) = f(b)$. Then $f(\{a\}\backslash \{b\}) = f(\{a\})\backslash f(\{b\}) = \{\}$, so $a$ must be $b$.