How to create a bijection between $\mathbb N$ and a set?

Question

Let the set $C = \mathbb N \times \{a,b\}$.

I wanna show that $C$ is countable but I'm so confused as to how or where I should start. Any tips/hints?

Answer 1

Hint: Start by listing the elements of $C$ one by one in a systematic manner that guarantees that any element is in the list somewhere. Then you're done.

Answer 2

If you find a bijection between a set and $\Bbb N$, it shows that the set is countable (by "counting" or "numbering" its elements).

In this case, an extremely simple bijection $f:\langle n\in\Bbb N, m\in\{a, b\}\rangle\to\Bbb N$ is: $$f(\langle n, m\rangle) = \begin{cases}2n-1 & m=a\\2n&m=b\end{cases}$$

This is a bijection because every natural number gets mapped to (it is "onto") and it maps from every natural number onto a single element (it is "one-to-one").

Answer 3

We have that $\Bbb{N} \times \{a,b\}=\bigcup_{n=1}^{\infty}\{(n,a)\} \cup \bigcup_{n=1}^{\infty}\{(n,b)\}$

So $\Bbb{N} \times \{a,b\}$ is a union of two countable sets thus countable.

that $\Bbb{N} \times \{a\}$ is countable using the bijection $f:\Bbb{N} \times \{a\} \to \Bbb{N}$ such that $f((n,a))=n$

With the same argument you can prove that $\Bbb{N} \times \{b\}$ is also countable.