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How do you represent a Pivot Table (as per Excel) as a mathematical mapping using set theory?

My assumption would be you're basically mapping a Cartesian Product to a count function. However, specifying the count function is where I'm unsure. This is my best attempt at doing one for a count pivot table (averages, sums etc. could be easily tweaked):

$$ C = \{a_i \in \text{Type 1}\} \times \{b_i \in \text{Type 2}\} $$

$$ Rows = \{r_i \in C\} $$

$$ C \mapsto \sum^{|Rows|}_{i=1} c()\text{, where } c() = \begin{cases} 1,& Row_i = (a_i, b_i)\\ 0,& Row_i \neq (a_i, b_i)\end{cases}$$

Is there a simpler way, or clearer more rigorous way?

How can you explicitly declareRows as a multiset?

Full disclosure, my university was pretty bad at providing a good set theoretic background. So my apologies.

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For each column in a table, there is some set it takes its values from. For example, one column could consist of arbitrary real numbers - the set $\Bbb R$, another column could be a fixed list of possible values of arbitrary type: $\{$East, West, South, North$\}, \{$Doris, Kirsten, Scot, Gerry, Lori$\}$, etc. Label the set for column number $c$ as $S_c$.

A Table of size $n$ subordinate to the set sequence $(S_c)_{c=1}^m$ is then a function $$T : \{1,2, ..., n\}\times\{1,2, ..., m\} \to \bigcup_{c=1}^m S_c$$ that satisfies the condition that for all $r$ and $c, T(r,c) \in S_c$.

Let $\mathscr T((S_c)_{c=1}^m)$ be the collection of all tables (of any size) subordinate to $(S_c)_{c=1}^m$. If $(\hat S_c)_{c=1}^k$ is another collection of sets, then a Pivot Table is simply a map $$PT: \mathscr T((S_c)_{c=1}^m) \to \mathscr T((\hat S_c)_{c=1}^k)$$

that is, it is some established method of converting a table subordinate to $(S_c)_{c=1}^m$ into a table subordinate to $(\hat S_c)_{c=1}^k$

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