## Quick Reference

The set of all subsets of a universal set E is closed under the binary operations∪(union) and∩(intersection) and the unary operation ′ (complementation). The following are some of the properties, or laws, that hold for subsets *A*, *B* and *C* of *E*:*A*∪(*B*∪*C*)=(*A*∪*B*)∪*C* and *A*∩(*B*∩*C*)=(*A*∩*B*)∩*C*, the associative properties.*A*∪*B*=*B*∪*A* and *A*∩*B*=*B*∩*A*, the commutative properties.*A*∪Ø=*A* and *A*∩Ø=Ø, where Ø is the empty set.*A*∪*E*=*E* and *A*∩*E*=*A*.*A*∪*A*=*A* and *A*∩*A*=*A*.*A*∩(*B*∪*C*)=(*A*∩*B*)∪(*A*∩*C*) and *A*∪(*B*∩*C*)=(*A*∪*B*)∩(*A*∪*C*), the distributive properties.*A*∪*A*′=*E* and *A*∩*A*′=Ø.*E*′=Ø and Ø′=*E*.*A*′)′=*A*.*A*∪*B*)′=*A*′∩*B*′ and (*A*∩*B*)′=*A*′∪*B*′, De Morgan's laws.The application of these laws to subsets of *E* is known as the algebra of sets. Despite some similarities with the algebra of numbers, there are important and striking differences.

*A*∪(*B*∪*C*)=(*A*∪*B*)∪*C* and *A*∩(*B*∩*C*)=(*A*∩*B*)∩*C*, the associative properties.

*A*∪*B*=*B*∪*A* and *A*∩*B*=*B*∩*A*, the commutative properties.

*A*∪Ø=*A* and *A*∩Ø=Ø, where Ø is the empty set.

*A*∪*E*=*E* and *A*∩*E*=*A*.

*A*∪*A*=*A* and *A*∩*A*=*A*.

*A*∩(*B*∪*C*)=(*A*∩*B*)∪(*A*∩*C*) and *A*∪(*B*∩*C*)=(*A*∪*B*)∩(*A*∪*C*), the distributive properties.

*A*∪*A*′=*E* and *A*∩*A*′=Ø.

*E*′=Ø and Ø′=*E*.

*A*′)′=*A*.

*A*∪*B*)′=*A*′∩*B*′ and (*A*∩*B*)′=*A*′∪*B*′, De Morgan's laws.

*Subjects:*
Mathematics.