The Property of Baire

Abstract

The operation of symmetric difference, defined by

$$A\Delta B = (A \cup B) - (A \cap B) = (A - B) \cup (B - A).$$

is commutative, associative, and satisfies the distributive law A ∩ (B △ C) = (A ∩ B) △ (A ∩ C). Evidently, A △ B ⊂ A ∪ B and $A\,\Delta \,A = \emptyset$. It is easy to verify that any class of sets that is closed under △ and ∩ is a commutative ring (in the algebraic sense) when these operations are taken to define addition and multiplication, respectively. Such a class is also closed under the operations of union and difference. It is therefore a ring of subsets of its union, as this term was defined in Chapter 3.