How to prove that if $A$ is infinite and $B$ is finite, then $|A\cup B|=|A|$?

Question

I'm studying logic and unfortunately, I'm a newbie at this, so I don't see the stuff everyone sees at the moment. I want to solve following exercise, but get nowhere:

Let $A$ be an infinite set and let $B$ be a finite set. Use the AC to prove that $|A| +|B| = |A|$.

I've proved this one though:

If $X$ is an infinite set, there is an injective function $\mathbb{N} \rightarrow X$, hence $\omega \leq |X|$.

But I don't see how I could use this. What's the logic here?

Answer 1

We can use your second theorem. Suppose that $f: \Bbb N \to A$ is an injection. Let $S = f(\Bbb N) \subset A$. Let $g:B \to \{1,\dots,|B|\}\subset \Bbb N$ be a map enumerating the elements of $B$.

We can now define the map $\phi:A \coprod B \to A$ by

$$ \phi(x) = \begin{cases} x & x \in A \setminus S\\ f(f^{-1}(x) + |B|) & x \in S\\ f(g(x)) & x \in B \end{cases} $$