Bijection of the set of natural numbers onto the set of integers.

Question

An example in Real Analysis by Sherbert and Bartle tells that the set of integers is a bijection of the set of natural numbers. How is the one to one correspondence possible for the set of integers? To be precise, give the bijection function that maps the sets of natural numbers to the set of integers?

Answer 1

Assuming natural numbers contain $0$. $$0\to0$$ if $x$ is an integer, $$x\to 2\lvert x \rvert - \frac 1 2 (1-\frac x {\lvert x \rvert})$$

Answer 2

Let me give you a description using a story first then we will go to the proof. This is associated with the Hilbert Grand Hotel Paradox. Suppose, you have a hotel having infinitely many room numbered $0,1,..., $ where all the rooms are occupied. A new group of people came at midnight consisting of infinitely many people numbered $-1, -2,...$ How can you place them in the hotel given that in a single room only one person can stay?

One possible way is leave the person in the $0$-th room and the $1$-st room as it was and ask all the existing persons to come out and shift all other persons two rooms after from where the last person went.. That sends the member in the second room to $3$-rd room, $3$-rd room to $5$-th room and so on, this makes the even rooms (starting from $2$ ) vacant. Now you can place the $-n$ numbered person to $2n$-th room. So the map is send $0 \rightarrow 0 , n \rightarrow 2n-1$ if $n>0 $ and $-n \rightarrow 2n$, where $n>0$. It gives you a bijection from $\mathbb Z$ to $\mathbb N \cup \{0\} $. Now you can shift them all to the right to one place to get the answer. So, the map is

$f(0) =1 , f(n) =2n ; n>0 , f(-n) = 2n+1 ; n>0 $

Answer 3

Take the set of naturals (with $0$). If odd, add $1$ and negate. Divide everything by $2$.

or

Take the set of naturals (without $0$). If odd, subtract $1$ and negate. Divide everything by $2$.

Answer 4

Instead of worrying about hard math just think about easy language.

We use the natural numbers to count with and to have a bijection from the counting numbers to a set is to arrange the set in order to count them.

Can you arrange the integers in order to count them?

That is to say. Can you list the integers one after another and count them as you go along?

I can list them $0,1,-1,2,-2,3,-3,......$

And I can count them: One, two, three, four, five, six, seven....

I can do that forever. So that's that.... That's a bijection..