If $A'$ and $B'$ have the same cardinality, as do $A\subseteq A'$ and $B\subseteq B'$, are $A'\setminus A$ and $B'\setminus B$ equipotent?

Question

If $A\subset A'$, $B\subset B'$, if $\operatorname{card}(A)=\operatorname{card}(B)$ and $\operatorname{card}(A')=\operatorname{card}(B')$, why is it that $\operatorname{card}(A'\backslash A)=\operatorname{card}(B'\backslash B)$ ?

Answer 1

It's not true. Consider $A'=B'=\mathbb N$, $A=\{x\in\mathbb N:x\ge3\}$, $B=\{x\in\mathbb N:x\ge9\}$.

Answer 2

Consider real interval $A=[0,2]$ of cardinality $\mathfrak C$.

We'll subtract four different intervals, non-degenerate hence each equipollent to $A$, and get distinct cardinalities of resulting sets:

$A \setminus [0,2] = \{\}$ – empty set, cardinality $0$.

$A \setminus [0,2) = \{ 2 \}$ – a singleton, cardinality $1$.

$A \setminus (0,2) = \{ 0, 2 \}$ – two-item set, cardinality $2$.

$A \setminus [1,2] = [0,1)$ – interval, cardinality $\mathfrak C$.