Finite sets, bijective functions This is a beginners question. I'm trying to understand the definition of a finite set. So I read that a set $A$ is finite if there exists a bijection $f: A \rightarrow \{1,...,n\},$ for some positive integer $n$. 
My understanding of a bijection is that it is a function where every element of one set is "paired" with an element of another set. So $f(x) = 2x$ is one example of a bijection.
So what if I have for example the set $A = \{1,3,4,7\}.$ Is that finite, because I can't think of a function that maps $A$ to $\{1,2,3,4\}?$
Or must a finite set be something like $B = \{5,10,15\},$ where $f(x) = x/5$ so $f$ maps $B$ to $\{1,2,3\}?$
If A is not a finite set, what is it then? Is there a third category (finite, infinite and then something else)?
 A: There is no rule that the function providing the bijection be "nice". So, in your example, we may just define a function $f$ whose domain is $\{1, 3, 4, 7\}$ as


*

*$f(1)=1$,

*$f(3)=2$,

*$f(4)=3$, 

*$f(7)=4$.
This provides the needed bijection. (Note that it's not the only option - we could also e.g. send $1$ to $4$, $3$ to $3$, $4$ to $2$, and $7$ to $1$.)
Crucially, note that this $f$ is defined only on the original set $\{1, 3, 4, 7\}$ - it makes no sense to ask what $f(0)$ is (say), nor should it. 

Incidentally, however, note that we could have done this with a polynomial via some appropriate interpolation. However, there's no actual reason to do this.
A: You are running into the usual beginner's pitfall of only thinking about functions as polynomials. (There was a time in my mathematical life when I did the same thing.)
But a function $f \colon A \to B$ is any valid rule specifying a unique value $f(a) \in B$ for each $a \in A$, it need not have a "nice" closed equational form.
That being said, you could also intuitively think about the bijection $f \colon A \to \lbrace 1, \dotsc, n \rbrace$ as counting the elements of your finite set $A$, i.e. the element that is mapped to $1$ is the first element of $A$, the element that is mapped to $2$ is the second, ..., and the element that is mapped to $n$ is the $n$-th and last element of $A$.
A: Your set $\{1,3,4,7 \}$ is finite. A bijection with the set $\{1,2,3,4\}$ would be the function $f$ that maps $1 \to 1$, $2 \to 3$, $3\to 4$ and $4\to 7$. The key here is that the range of the functions can be defined for each element separately. 
