Explicit Definitions of Functions that aren't on the real numbers or built with objects that are on the real numbers I am left perplexed by how arbitrary maps between arbitrary sets are practically implemented and defined explicitly without literally listing long tables of maps $(a,b)$. 
Suppose $f$ is a function. Then $f$ is a set $\{(a,b)\}$. For real numbers or any object built with real numbers, we can write $f$ using expressions involving operations on these real objects or objects derived from reals instead of listing out the set explicitly. 
However for those non real or real derived sets, how are functions defined explicitly. I am writing all these theorems about bijections and images of functions but cannot fathom how non-real functions can be defined explicitly as an operation as opposed to a long list. 
Can I define operations on non-real or real derived sets (eg: $R^n$ is real derived, $C$ is real derived
but many objects in discrete math/computer science do not involve reals)
 A: Example of a function:
Let $W$ be the set of all words in the letters $a, b, c$.
$W = \{a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, ..., abcbabacbaabba, ...\}$
Define the function C (called 'concatenation')  on $W \times W$ into $W$ as follows:
$(u,v) \mapsto uv$
i.e. for two 'words' $u$ and $v$ in $W$, stick $v$ onto the right of $u$.

Mathematics entails meaningful discourse within a bigger there-for-grabs-math-world playground.
The number of known atoms in universe is less than $10^{100}$. The finite set
$\{ 1, 2, 3, ..., 10^{100}\}$
is an example of a set with exactly that many elements. Is it practical to consider an arbitrary function of this set into itself?
This quote might help:
Mathematics is a game played according to certain simple rules with meaningless marks on paper. -David Hilbert
So, if you wanted to, you could insist that every function have a 'practical' description. Just define what practical means in your math play ground.
You might be interested in a philosophy of mathematics called Constructivism and another one called Finitismn
A: When talking about a specific function you only have to make clear what this function is supposed to do. When I say to you, that $f(x)$ takes in a shoe $x$ and spills out the other shoe for it (the left one for a right shoe $x$ and vice versa), then this is a valid description of a process and you fully understand what it does. There is no need to use formulas and numbers. And you must admit, that the set of shoes is rather far away from a real ($\Bbb R$) set (or real derived set as you would call it).
I can do rather obscure stuff on numbers too. For example define the function $f(x)$ that maps a rational number $\frac pq$ to zero, but any irrational number is left untouched. I made the process clear to you. But good luck describing this with an elementary formula.
Real sets are also just sets with elements and like with any other set we can shuffle their elements. This is what functions do. Shuffle elements in a more or less regular way (yes I know not all functions are bijective). Some nice and/or useful forms of shuffeling are denoted by us with some shorthand formulas like $x+1$ or $x^2$.
A: You've really hit the nail on the head. For an arbitrary function between two sets, you will have to list out the list of ordered pairs $(a,b)$. Of course for certain special functions you may have an equation, formula, or rule that can be described in plain words. But there are also often (depending on the sets) many more functions that cannot be described in this way.
Take for example an arbitrary function from $\mathbb{R} \to \mathbb{R}$. Some functions can be described with the standard operations on real numbers, but it is completely impossible to describe every function this way. And of course it is also impossible to list out the ordered pairs $(a,b)$ describing such a function. So you are left with infinitely many functions $f: \mathbb{R} \to \mathbb{R}$ which cannot be described in any way.
For sets that are not on the real numbers, if your function arises in a natural way you can maybe describe your function using words. Or you can translate your objects into numbers, and maybe that will allow you to write a formula (for example if your set has $7$ objects, consider it to be the set of numbers $\{0,1,2,3,4,5,6\}$). But for arbitrary functions, listing the ordered pairs will be the only guaranteed way to represent it.
A: We don't need to know what a function does to know that a function exists and to analyze it.  And, just to confound you, real valued functions don't need to have operations or definitions.  I know there is an utterly undefinable function $f$ so that $f(\sqrt{e})=27.6$ and if $w =  "\sqrt{e}$ with its 27th digit 389th digits exchanged" then $f(w)$ is some real number that has no sensible way to describe, and that every real number is mapped to another in no predictable or knowable manner.  I can not know this function in any way and the universe doesn't have enough lifetimes to list it.  But that doesn't matter.  I know it must exist.
Why must it exist?  Because all potential functions exist. How can I reasonable refer to "that" function and given another function and say, no that's a different one?  Or how can I reasonably tell you this is "my" function and have you have any idea which one I mean?  (theideasmith: "What is f(57.1)?"  fleablood:"point one followed by one zero then one followed by two zeros and so on except every marcenne prime over 32 place digit is replaced with 7.  What is f(6)?" theideasmith: "I don't know $\pi^e$?" fleablood: "Nope. It's 8. You don't get this function at all, do you?")  You can't really.  But there's no reason we have to be able to know everything about something to talk about it. 
Let me draw an analogy: "Let $p$ be a prime number larger than   $10^{(10^{100})}$". How is it possible that I can make such a statement?  I have not stated what the number is precisely and the only way to do so would be to list all the digits.  But it has at least a googol digits and... I simply can't do that.  So does that mean I can not make such a statement?
Why are functions any different than numbers.
