Number of Nonnegative integers = number of integers I just came across the idea that says the number of non-negative integers is equal to the number of integers. How is this possible? Isn't it true that non-negative integers are a subset of all integers? Then, it should follow that the number of all integers is bigger, twice at least, than the number of non-negative integers?
If the passage is indeed true, what is the implication of this?
 A: The idea here is that if we can put a set into a one-to-one correspondence with the Natural Numbers, then the sets have the same cardinality $\aleph_0$. 
So the number of nonnegative integers (we can also call this the cardinality of the set $\mathbb{N}$) is the same as the number of integers (we can call this the cardinality of $\mathbb{Z}$). We can show that $\mathbb{Z}$ has cardinality $\aleph_0$ because the function $f:\mathbb{Z}\rightarrow \mathbb{N}$ given by:
$$f(z) = \begin{cases} 2z + 2 \quad\quad \;if\;z\geq0\\-2z-1, \quad if\; z<0\end{cases}$$
is a bijection.[*]
This can be a little tricky to wrap your head around - if you think that it's not possible because both sets "go to infinity" and one of them seems "smaller" than the other, remember that infinity is not a number.

[*] The function described there maps from $\mathbb{Z}$ to $\mathbb{N}$, where $\mathbb{N}$ is defined as the positive integers. Given that the OP asked for the nonnegative integers, i.e. $\mathbb{N}$ indexed from $0$, the function is not perfectly applicable, but the idea remains the same, and modifying the function to map from $\mathbb{Z}$ to the nonnegative integers is easy, perhaps best left as an exercise for OP.
