A set of all the natural numbers can biject to a set of all the even numbers? Is this a disprove? I saw that thing that if you biject set of all the natural numbers and a set of all the even numbers, they are able to pair.
I don't think you can compare two infinites just like this I believe there's still a way to solve it better.
My solution is:
Let $E$ be all the even numbers. Let $O$ be all the odd numbers. If you put together the even numbers and the odd numbers, this is called the natural numbers.
Now what need to do is biject between $E\cup O$ and $E$, what that happens is that $E$ is removed and what is left is $O$ and nothing. Isn't it means that they don't biject?
What I want to know is that why I am wrong.
 A: You set up is: $E$ is the set of even numbers, $O$ is the set of odd numbers, and $E\cup O$ (that is, all the members of either $E$ or $O$) is just the set of all natural numbers.
The task is to find a bijection between $E\cup O$ and $E$. When we do this, we end up with $O$ and $\{\}$ (presumably the thinking is that we've taken $E$ away from each).
What this shows is that a very natural function between the two sets – identity – is not a bijection. That is, when we map $f:E\to E\cup O$ with $n\mapsto n$, the image of the function is just $E$ – we miss out 'half' of the target set.
But for the sets to be the same size, we don't require that every function between them is a bijection. We just require that some function between them is a bijection. We can still use the standard example of a bijection, where $n\mapsto n/2$, which is a bijection.
A: Here is a bijection between $\{0,1,2,3,\ldots\}$ and the subset of all even numbers in that set:
\begin{align}
0 & \longleftrightarrow 0 \\
1 & \longleftrightarrow 2 \\
2 & \longleftrightarrow 4 \\
3 & \longleftrightarrow 6 \\
4 & \longleftrightarrow 8 \\
5 & \longleftrightarrow 10 \\
& {}\quad\vdots
\end{align}
If you think of any even number, I know that there will be exactly one number in the left column that it corresponds to, even if I don't know which even number you picked.  And if you think of any number in the left column, I know that there will be exactly one number in the right column that it corresponds to, even if I don't know which number you picked.
One of the difference between finite and infinite sets is that there can be a bijection between an infinite set and a proper subset of itself.
