[Math] Prove: if $A$ is infinite set and $B$ is finite set then $\left| {A – B} \right| = \left| A \right|$

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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.

Best Answer

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.

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