# 4.2 The Boolean Algebra of Sets

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It is clear that set theory is closely related to logic. The intersection and union of sets can be defined in terms of the logical ‘and’ and logical ‘or’ operators. The notation {*x *| *P*(*x*)} makes it possible to use predicates to specify sets. And if *A *is any set, then the formula *x** *∈ *A *defines a one place predicate that is true for an entity *x *if and only if *x *is a member of *A*. So it should not be a surprise that many of the rules of logic have analogues in set theory.

For example, we have already noted that ∪ and ∩ are commutative operations. This fact can be verified using the rules of logic. Let *A *and *B *be sets. According to the definition of equality of sets, we can show that *A*∪*B *= *B*∪*A *by showing that ∀*x*((*x *∈*A *∪ *B*) ↔ (*x *∈ *B *∪ *A*)). But for any *x*,

*x *∈ *A *∪ *B *↔ *x *∈ *A *∨ *x *∈ *B *(definition of ∪)

↔ *x *∈ *B *∨ *x *∈ *A *(commutativity of ∨)

↔ *x *∈ *B *∪ *A *(definition of ∪)

The commutativity of ∩ follows in the same way from the definition of ∩ in terms of ∧and the commutativity of ∧, and a similar argument shows that union and intersection are associative operations.

The distributive laws for propositional logic give rise to two similar rules in set theory. Let *A*, *B*, and *C *be any sets. Then

*A*∪(*B*∩*C*) = (*A*∪*B*)∩(*A*∪*C*)

and

*A*∩(*B*∪*C*) = (*A*∩*B*)∪(*A*∩*C*)

These rules are called the distributive laws for set theory. To verify the first of these laws, we just have to note that for any *x*,

*x *∈ *A*∪(*B*∩*C*)

↔ (*x *∈ *A*) ∨ ((*x *∈ *B*) ∧ (*x *∈ *C*)) (definition of ∪, ∩)

↔ ((*x *∈ *A*) ∨ (*x *∈ *B*)) ∧ ((*x *∈ *A*) ∨ (*x *∈ *C*)) (distributivity of ∨)

↔ (*x *∈ *A *∪ *B*) ∧ (*x *∈ *A *∪ *C*) (definition of ∪)

↔ *x *∈ ((*A *∪ *B*) ∩ (*A *∪ *C*)) (definition of ∩)

The second distributive law for sets follows in exactly the same way.