Example Infinite set and a bijection from the set I was wondering if someone could give me an example of an infinite set and a bijection from the set to a proper subset of itself. 
 A: $$f: \mathbb N\rightarrow 2\mathbb N$$
$$n \mapsto 2n$$
It's a bijection from the set of all natural numbers $\{1, 2, 3, \cdots\}$ into its proper subset of all even natural numbers $\{2, 4, 6, \cdots\}$.
A: One example was given in the comments.  Here’s another:  
$$n \mapsto n+1$$
is a bijection from {$0,1,2,...$} to {$1,2,...$} $\subsetneq $ {$0,1,2,...$}.
A: This is easy and more general than you think.
If $f:A \to A$ is one to one then $f:A\to f(A)\subseteq A$ is onto and a bijection.  If $f:A \to A$ is not onto then $f:A \to f(A)\subsetneq A$ is a bijection to a proper subset.
So all we need is 1) An infinite set $A$.  2) An injective but not onto function from $A\to A$.
$f:\mathbb R \to \mathbb R$ via $f(x) = e^x$ is one to one and will always have $e^x > 0$ and the image of $f$ is $(0,\infty)$.
So $f:\mathbb R \to (0,\infty)$ via $f(x) = e^x$ is an example.
But probably the very first one most of us have probably heard of is:
"You know, there are the same number of even numbers and numbers because for every number if you multiply it by $2$ you get an even number."  
Most of us heard that in the first grade or so (and lost three weeks of sleep immediately afterward).
This is $f: \mathbb N \to \{$even integers$\}$ via $f(n) = 2n$ is such a bijection.
Less easy but maybe more fundamental and more to the point is:
we know $|\mathbb Q| = |\mathbb Z| = |\mathbb N|$ so by definition we know there are bijections from $\mathbb Q \to \mathbb Z$ and from $\mathbb Q \to \mathbb N$ and from $\mathbb Z \to \mathbb N$.
So we can say if $q\in \mathbb Q^+$ and $q = \frac ab > 0$ in lowest terms and we write down all the pairs of integers in the following order $(1,1),(2,1),(1,2),(3,1),(2,2),(1,3), (4,1),(3,2),(2,3),(1,4),(5,1),(4,2),.....$ and we cross out all the pairs that are not relatively prime so we get $(1,1),(2,1),(1,2),(3,1),(1,3),(4,1),(3,2),(2,3),(1,4),(5,1),(1,5),(6,1)....$ and count where $(a,b)$ appears in the list... then that is a bijection from $\mathbb Q^+ \to \mathbb N$.
