I am new to the topic of cardinality and I am trying to prove the following statement:
"If $a$ is a natural number then $\mathbb{N} \setminus \{ a \}$ is denumerable. Here, $\mathbb{N} \setminus \{ a \}$ is the set $\mathbb{N}$ with the number $a$ removed".
I understand that a set is denumerable if it has the same cardinality as the natural numbers and that one way to show this is to prove that there is a bijection from the natural numbers to the set in question. However, I am having a hard time coming up with such a function that is a map. Are there specific steps for choosing a function that maps two sets?
For example. I've tried to follow the proof that $(0, 1)$ and $(1, \infty)$ have the same cardinality. The function that is used to map $(0, 1)$ to $(1, \infty)$ is $1 \over x$. How was this chosen?
Define $f(n)=\begin{cases}~~n\quad\quad \text{if} \quad n<a\\n-1\quad\text{if}\quad n> a. \end{cases}$