I came up with a way to do set theory using polynomial arthimetic.
To explain, consider two polynomials $P(x)$ and $Q(x)$, then we can observe the following analogies:
1. In Polynomial addition
$$ |P(x)|+|Q(x)|$$
Is analogous to set intersection of sets because the above equation is only true for $x$ values which make both $P$ and $Q$ zero at the same time.
2. In Polynomial multiplication
$$|P(x)| \cdot |Q(x)|$$
Can be thought of as set union because the above equation is true for the numbers which satisfy either $P$ or $Q$
3. In polynomial division
$$ \frac{|P|}{|Q|} $$
The above expression can be thought of a set difference, we remove the zeros of Q which exist also in P.
Analogies to set complements:
Polynomial reciprocal is similar to set inverses, for example
$$ |P(x)^{-1}| = \frac{1}{|P(x)|}$$
Is defined for all values of $x$ except where $P(x)=0$ where it is undefined. To get the whole universal set back, we multiply inverse with regular one:
$$|P(x)| \cdot \frac{1}{|P(x)| }=1$$
Proving set identities using polynomials
- $ A \cup (B \cup C)= (A \cup B) \cup C$
This is directly analogous to associativity of polynomial multiplication , considering three polynomials $P,Q,R$:
$$ ( |P| \cdot |Q|) |R| = |P|( |Q| \cdot |R|)$$
- $A \cup B = B \cup A, A \cap B , B \cap A$
This is directly analogous to communality of polynomial multiplication.
Demorgan's laws
- $$ A^c \cup B^c = (A \cup B)^c$$
Easy relation to understand with polynomial way, consider two polynomials $P(x),Q(x)$, then it is trivial that:
$$ \frac{1}{|P|} \cdot \frac{1}{|Q|} = (|PQ|)^{-1}$$
- $$ (A \cap B)^c= A^c \cup B^c$$
This one just turns out weird, but it works well for most identities where there isn't addition and multiplication on both sides
My question: Has this idea been studied before? Where there any new insights from thinking of sets like this?
[Note: I know of generating functions already]
Best Answer
A rather elaborate theory related to this was worked out over a period of several decades by the Russian mathematician Vladimir Logvinovich Rvachev (1926-2006) (biography), called called $R$-functions. See Semi-analytic geometry with R-functions by Vadim Shapiro (2007; footnote on p. 3 discusses the origin of the name "R-function").