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Set identities

We now consider the basic set identities that relate the various set operations.

The sets A, B, C below are subsets of a universal set U.

Identity Laws

A∪∅=A,A∩U=A

Domination Laws

A∪U=U,A∩∅=∅

Idempotent Laws

A∪A=A,A∩A=A

Complement Laws

A∪Ac=U,A∩Ac=∅

Double Complement Law

(Ac)c=A

Commutative Laws

A∪B=B∪A,A∩B=B∩A

Associative Laws

A∪(B∪C)=(A∪B)∪C,A∩(B∩C)=(A∩B)∩C

Distributive Laws

A∪(B∩C)=(A∪B)∩(A∪C),A∩(B∪C)=(A∩B)∪(A∩C)

De Morgan's Laws

(A∪B)c=Ac∩Bc,(A∩B)c=Ac∪Bc

Absorption Laws

A∪(A∩B)=A,A∩(A∪B)=A

Complements of U and âˆ…

Uc=∅,∅c=U

Set Difference Law

A∖B=A∩Bc


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