Construction of Sets

Intersection

Let and be two given sets. The intersection is the set:
More generally, we can intersect a collection of sets: , where is some index set, and are sets.

Union

The union of sets and is the set:
More generally, we can union a collection of sets: , where is some index set, and are sets.

Disjoint Union

The disjoint union of and is the set:

Difference and Complement

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 as
where each is an ordered -tuple and the is called the -th coordinate.
In general, we can also define possibly infinite Cartesian product of sets. Suppose is an index set, is a family of sets. Then we define

Theorem

The cartesian product satisfies the following universal property: Let be a family of sets and be a set. Then for any family of maps , there exists a unique map such that for all , where is the projection map defined by for all .

Proposition

Suppose and are families of sets for arbitrary index sets and . Then

Proof For any element , say and . Then . The converse is similar.