Let $A_1$, $A_2$, $B_1$ and $B_2$ be sets such that
$$
|A_1|=|A_2|,\quad|B_1|=|B_2|,\quad B_1\subset A_1,\quad B_2\subset A_2.
$$
Where $|\ \ |$ denotes the cardinality of a set.
How to show that $|A_1\setminus B_1|=|A_2\setminus B_2|$ ?
We know that there is a bijection from $A_1$ to $A_2$ and one from $B_1$ to $B_2$, and we just have to find one from $A_1\setminus B_1$ to $A_2\setminus B_2$, but how to prove its existence?
Best Answer
But this is not true at all...
Consider the following example:
$A_1=A_2=\mathbb N$ and $B_1=\{n\in\mathbb N\mid n\text{ is even}\}$, $B_2=\{n\in\mathbb N\mid n>3\}$.
Clearly all sets involved have the same size but $A_1\setminus B_1$ is infinite where as $A_2\setminus B_2$ is finite.