The Property of Baire
Part of the Graduate Texts in Mathematics book series (GTM, volume 2)
The operation of symmetric difference, defined by
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.
$$A\Delta B = (A \cup B) - (A \cap B) = (A - B) \cup (B - A).$$
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© Springer-Verlag New York 1971