Does there exist an injection $\mathbb{R} \to\mathbb{N} \times \mathbb{ Z} \times \mathbb{ Q}$?

Question

I'm thinking since $|\mathbb{R}|>|\mathbb{N}| \times| \mathbb{ Z} |\times |\mathbb{ Q}|$ then there can't be an injection, but is that true?

Answer 1

Define $N:=\mathbb{N}\times \mathbb{Z}\times \mathbb{Q}$. Assume there exists such an injection $f: \mathbb{R} \rightarrow N$. Then the function $g: \mathbb{R} \rightarrow f(\mathbb{R}), g(x)=f(x)$ is a bijection (check this). As $N$ is countable (as it is the cartesian product of finitely many countable sets) and $g$ is a bijection, we would get that $\mathbb{R}$ is countable as well, which gives the desired contradiction.