Prove: if $A$ is infinite set and $B$ is finite set then $\left| {A - B} \right| = \left| A \right|$

Question

Prove: if $A$ is infinite set and $B$ is finite set then $\left| {A - B} \right| = \left| A \right|$

Well, one way to show it, is to find an injective function, for both directions.

First, Lets define: $C = A-B$.

$f:A\rightarrow C$ such that $f(x) = x$
$g:C\rightarrow A$ such that $g(x) = max(B) + x$

Hence,
$$\left| C \right| = \left| {A - B} \right| = \left| A \right|$$

I have three questions:

1. Is my solution right?
2. Assuming I am right, Is there a good alternative for $g(x)$? I don't like the idea of using the $max$ function in this kind proof.
3. Is there another solution other then finding two injective functions?

Thanks in advance.

Answer 1

Your solution is wrong, because it assumes that there is some sort of additive structure and an order defined on $A$ (by taking $\max B$ and $+$ into account).

This is true if $A$ is a set of integers, or real numbers, but not in general.

Instead try proving something else, the opposite, if $A$ is infinite and $B$ is finite, then $|A\cup B|=|A|$. This is slightly simpler, and you can reduce to the case where $A$ and $B$ are in fact a sets of integers.