A hint for a simple question about cardinality.

Question

Somebody can to give me a hint in the following question?

Let $X$ be an infinite set. Show that $X$ has the same cardinality that $X \cup \mathbb{Q}$.

Answer 1

Hint: Since $X$ is infinite we can write it as $X'\cup Y$ where $X'\cap Y=\varnothing$ and $|Y|=|\Bbb N|$. Since $|\Bbb N|=|\Bbb N\times\{0,1\}|$, we can use this to find a bijection between $Y$ and $Y\cup\Bbb Q$.