While going through my professor' script, I have found the following statement without proof:
If $X$ is a infinite set and $Y$ is at most countable then $\lvert X \rvert = \lvert X \cup Y\rvert$.
Can someone give me a hint or proof for this?
$X$ is infinite and $Y$ at most countable. Why is $\lvert X \rvert = \lvert X \cup Y \rvert$?
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cardinals
1 Answers
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If $ X $ is infinite, by countable choice, there's an injection $ \mathbb N \to X $, which allows us to realize it as a subset of $ X $. Then, $ X = \mathbb N \cup (X - \mathbb N) $ where the union is disjoint. Now, note that
$$ X \cup Y = \mathbb N \cup (X - \mathbb N) \cup Y = (\mathbb N \cup Y) \cup (X - \mathbb N) \cong \mathbb N \cup (X - \mathbb N) = X $$
where the fact that $ Y $ is at most countable implies that there's a bijection $ \mathbb N \to \mathbb N \cup Y $.