Operation on sets §

Strarting from fondamental $\in$ relation and the axiomns of ZFC Axioms of set theory we have defined some relation/operation on sets. Our goal here is to find the useful relations beetween them.

Siano $A,B,C$ insiemi.

Unions and intersections are associative: §

$$(A\cup B)\cup C=A\cup (B\cup C)$$ $$(A\cap B)\cap C=A\cap (B\cap C)$$

They are mutually distributive: §

$$(A\cup B)\cap C=(A\cap C)\cup (B\cap C)$$ $$(A\cap B)\cup C=(A\cup C)\cap (B\cup C)$$

Here we define an operation usefull in mesure theory, the symmetric difference

$$A\Delta B:=(A\setminus B)\cup (B\setminus A)$$

Equivalences §

$$A\cup B=(A\Delta B)\Delta (A\cap B)$$ $$A\setminus B=A\Delta (A\cap B)$$