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?

  • $\begingroup$ That's three questions... :-). $\endgroup$ – copper.hat Dec 19 '15 at 16:56
  • $\begingroup$ In general, there are no specific steps. Sometimes a 'natural' bijection is clear, sometimes not. Often one is clear but hard to write down concisely. Regarding the last problem, there may be many bijections (for example, $x \mapsto {1 \over 1-x}$ is another), so there is no unique way. After a while your bag of tricks will suggest possibilities. $\endgroup$ – copper.hat Dec 19 '15 at 17:02

Define $f(n)=\begin{cases}~~n\quad\quad \text{if} \quad n<a\\n-1\quad\text{if}\quad n> a. \end{cases}$

  • $\begingroup$ How did you choose this? $\endgroup$ – dreamin Dec 19 '15 at 17:01
  • $\begingroup$ @dreamin Arrange elements of $\mathbb{N}-\{a\}$ in increasing order, that is the bijection. $\endgroup$ – Suhail Dec 19 '15 at 17:03

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