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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?

tetris11
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    So the input takes two elements, and the output is one element? This is really unclear. Can you maybe give an example of just one specific mapping from one specific input to one specific output? – Bram28 Oct 24 '16 at 13:25
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    This seems like just a cartesian product. Given set $A$ and set $B$, the cartesian product is just the ordered pairs $(a,b)$ where $a\in A$ and $b\in B$. Are you wanting unordered pairs (i.e. sets of size 2)? Or what exactly? – Justin Benfield Oct 24 '16 at 13:28
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    It sounds like maybe you want $A \times B \mapsto A \times B$, with $A = { c_0,c_1 }$ and $B = { c_2,c_3 }$? – Bram28 Oct 24 '16 at 13:30
  • @Bram28 - added more examples – tetris11 Oct 24 '16 at 13:32
  • @JustinBenfield - sorry math is not at all my strong point - yes, I'm trying to emphasize that the output ordering is important and that the sets should be exactly of size 2. – tetris11 Oct 24 '16 at 13:34
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    Ok, question for you is the resulting (ouput) pair, $(c_i,c_j)$ ordered? Is the input any element from the set $A\times B$ (defined as Bram28 did)? – Justin Benfield Oct 24 '16 at 13:34
  • @JustinBenfield - Ah I see - No, the output is not ordered, and.... for $A \times B$ valid output would be ${c_0,c_2}$ or ${c_2,c_0}$ or ${c_1,c_2}$ or ${c_2,c_1}$ or ${c_0,c_3}$ or ${c_3,c_0}$. – tetris11 Oct 24 '16 at 13:37
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    What about inputs?, are they ordered? e.g. is $(c_0,c_1)$ different from $(c_1,c_0)$? What would the valid list of inputs look like? – Justin Benfield Oct 24 '16 at 13:41
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    I am still really confused what kind of mapping you want. Do you have a context for this? A word problem for which you are trying to use this formalization? – Bram28 Oct 24 '16 at 13:56
  • @JustinBenfield - $(c_0,c_1) =/= (c_1,c_0)$. Valid inputs would be $(c_0,c_1),(c_2,c_3)$ or $(c_1,c_0),(c_2,c_3)$ or $(c_0,c_1),(c_3,c_2)$. – tetris11 Oct 24 '16 at 14:07
  • @Bram28 - Yeah sorry - I'm trying to represent standard mendelian inheritance of parental alleles. Parents have two alleles each, and for a given locus the child inherits one from each without any attention to the ordering. – tetris11 Oct 24 '16 at 14:09
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    That situation I would think would correspond to a cartesian product, you can impose an (arbitrary) ordering on the two alleles in order to get the order structure needed to define a cartesian product. I remember doing punnett squares in biology in high school. (e.g. https://en.wikipedia.org/wiki/Punnett_square) – Justin Benfield Oct 24 '16 at 14:21

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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

$f:X\times Y\rightarrow A\square B$

Justin Benfield
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