Cardinality of the union of infinite and countable sets

This seems evident, but I cannot come up with a reasonable proof for:

Question: show that if $X$ is an infinite set and $Y$ is a countable set, then $|X \cup Y|=|X|$

Solution 1:

An Idea: Suppose first that $X$ and $Y$ are both countable, say $X = \{x_1, x_2, ...\}$ and $Y= \{y_1, y_2, ...\}$. Then the function $\phi : Y \to X \cup Y$ defined by $$\phi(y_i) = \begin{cases} x_{i/2} & i \text{ even} \\ y_{(i+1)/2} & i \text{ odd} \end{cases}$$ is a bijection, and therefore $|X \cup Y| = |Y|$. More generally, if $Y$ is infinite, there is a countable subset $\tilde{Y} \subset Y$. From what we just showed, you know that there is a bijection $\varphi : X \cup \tilde{Y} \to \tilde{Y}$. Extend this to a bijection $\Phi : X \cup Y \to Y$ by $$ \Phi(z) = \begin{cases} \varphi(z) & z \in X \cup \tilde{Y}\\ z & z \in Y \setminus (X \cup \tilde{Y}) \end{cases} $$

Solution 2:

HINT: First show that every infinite set $X$ can be written as the disjoint union of $X_0$ and $X_1$ with $X_0$ being countably infinite. Next show that the union of two countable sets is countable.