Prove that if
- $A\cap B\subsetneq A\cap C$, and
- there is an injective function between $A\setminus C$ and $A\setminus B,$
then $A\setminus B$ is an infinite set.
I know that if I can prove $A\setminus B\supsetneq A\setminus C$ I'm done but I'm just struggling to do it.
I've tried to proof it with definitions of intersection and relative complement but didn't succeed. I've tried to prove it also by contradiction but didn't succeed also.
The statement is false as written. Take $A=\{1\}$, $B=C=\{2\}$. Then $A\cap B=A\cap C=\emptyset$ and the identity function maps $A\setminus C=A$ to $A\setminus B=A$. However, $A\setminus A$ is not an infinite set.
If $\subset$ means strict subset, then take $A=C=\{1\}$ and $B=\{2\}$, and the same problem arises. The empty function maps $A\setminus C=\emptyset$ to $A\setminus B=B$.