Set Operations and the Laws of Set Theory
•The union of sets Aand Bis the set A∪B={x:x∈A∨x∈B}.
•The intersection of sets Aand Bis the set A∩B={x:x∈A∧x∈B}.
•The set difference of Aand Bis the set A\B={x:x∈A∧x6∈ B}.
Alternate notation: A−B.
•The symmetric difference of Aand Bis A⊕B= (A\B)∪(B\A).
Note: A⊕B={x: (x∈A∧x6∈ B)∨(x∈B∧x6∈ A)}.
The universe,U, is the collection of all objects that can occur as elements of the sets under
consideration.
•The complement of Ais Ac=U \ A={x:x6∈ A}.
For each Law of Logic, there is a corresponding Law of Set Theory.
•Commutative: A∪B=B∪A, A ∩B=B∩A.
•Associative: A∪(B∪C)=(A∪B)∪C, A ∩(B∩C)=(A∩B)∩C
•Distributive: A∪(B∩C) = (A∪B)∩(A∪C), A ∩(B∪C) = (A∩B)∪(A∩C)
and also on the right: (B∩C)∪A= (B∪A)∩(C∪A),(B∪C)∩A= (B∩A)∪(C∩A)
•Double Complement: (Ac)c=A
•DeMorgan’s Laws: (A∪B)c=Ac∩Bc,(A∩B)c=Ac∪Bc
•Identity: ∅ ∪ A=A, U ∩ A=A
•Idempotence: A∪A=A, A ∩A=A
•Dominance: A∪ U =U, A ∩ ∅ =∅
Arguments that prove logical equivalences can be directly translated into arguments that prove set
equalities.
Set equalities of note:
•A\B=A∩Bc
•A⊕B= (A∪B)\(A∩B)
