I have elements $\{c_0, c_1\}$ from one input set, and elements $\{c_2, c_3\}$ from another input set.
Each input can be in any order, and both inputs together (through some function) produce output which is any permutation of the input element of one set, with an input element of the other.
The closest I've come up with is:
$\{\{[c_0,c_1]\},\{[c_2,c_3]\}\} \mapsto \{\{[c_0,c_2]\},\{[c_0,c_3]\},\{[c_1,c_2]\},\{[c_1,c_3]\}\}$
But I'm not sure if this is proper notation, and I'm also not sure if it conveys that the output should be only one of the elements in the output set.
e.g.1 $(c_1,c_0),(c_3,c_2) \mapsto (c_2,c_0) $
e.g.2 $(c_0,c_1), (c_2,c_3) \mapsto (c_0,c_3) $
e.g.3 $(c_1,c_0), (c_2,c_3) \mapsto (c_0,c_3) $
(where order of elements is deliberate)
How do I best represent this mathematically?
So if I understand correctly what your situation looks like, you have inputs which are ordered pairs of ordered pairs.
Specifically, you taking inputs from the set $X\times Y$, where $X=\{(c_0,c_1),(c_1,c_0)\}$ and $Y=\{(c_2,c_3),(c_3,c_2)\}$. The outputs are sets of size 2 where the one element comes from the set $A=\{c_0,c_1\}$ and the other element comes from $B=\{c_2,c_3\}$. As far as I know, this set has no standard notation, so you can define your own, I will use
$A\square B:=\{\{c_0,c_2\},\{c_0,c_3\},\{c_1,c_2\},\{c_1,c_3\}\}$
for this set. So now, your situation can be described as having a function, $f$, from $X\times Y$ to $A\square B$, which can be expressed in notation as
$f:X\times Y\rightarrow A\square B$