Construction of Sets

Intersection

Let and be two given sets. The intersection is the set:

Union

The union of sets and is the set:

Disjoint Union

The disjoint union of and is the set:

Difference

The difference of from is the set In the circumstance that sets are considered within a fixed set , for any subset of we write as and call it the complement of in .

Proposition

Let , , and be sets. The following properties hold:

• Commutativity: , .
• Associativity: , .
• Distributivity: , .
• de Morgan’s laws: , .

Proof These are direct consequences of the arithmetic of logical connectives.

Cartesian Product

Cartesian Product

Let be sets. The setis called the Cartesian product of and is written aswhere each is an ordered -tuple and the is called the -th coordinate.