Set Identities



Definitions:

Universal set : I

Empty set: 

Union of sets

AB={x:xA  or  xB}

Intersection of sets

AB={x:xA  and  xB}

Complement

A={xI:xA}

Difference of sets

AB={x:xA  and  xB}

Cartesian product

A×B={(x,y):xA  and  yB}

Set identities involving union

Commutativity

AB=BA

Associativity

A(BC)=(AB)C

Idempotency

AA=A

Set identities involving intersection

Commutativity

AB=BA

Associativity

A(BC)=(AB)C

Idempotency

AA=A

Set identities involving union and intersection

Distributivity

A(BC)=(AB)(AC)
A(BC)=(AB)(AC)

Domination

A=
AI=I

Identity

A=
AI=A

Set identities involving union, intersection and complement

Complement of intersection and union

AA=I
AA=

De Morgan’s laws

(AB)=AB 
(AB)=AB 

Set identities involving difference

BA=B(AB)
BA=BA
AA=
(AB)C=(AC)(BC)
A=IA