Sets

Sets are a collection of members.

Description

A set is a collection of members. Generally, sets are defined and notated like {a,b,c}.

A set can also be defined by a condition. For example, {x ∈ R | x > 0} should be read as the set of all real values greater than 0.

Sets carry a primitive concept of size in the form of cardinality: they can be countable (i.e., there are n members), countably infinite (i.e., the set can be mapped to the natural numbers), or uncountable. More complex concepts of size rely on measures.

A set may or may not be ordered.

If a set is closed on addition and multiplication, it is actually then a field.

Openness of a set can be determined in either a topological space or a metric space. The core idea is that a set is closed if it contains all of its limit points, and open if it contains none of them.

Commonly Used Sets

The empty set is notated Ø.

The commonly re-used notations for basic sets of numbers are:

• Natural numbers (N)

• Integers (Z)

• Rational numbers (Q)

• Real numbers (R)

Logic

Membership

a ∈ A means a is a member of A. Conversely, a ∉ A means a is not a member of A.

Subsets

A ⊆ B means that A is a subset of B. ∀ a ((a ∈ A) -> (a ∈ B)).

This leads to a test for set equality: if A ⊆ B and B ⊆ A.

If A ⊆ B and A != B, then A is a proper subset of B. This is notated as either A ⊂ B or A ⊊ B.

The reverse relation could be notated as B ⊇ A, B ⊃ A, and so on.

Complements

The complement of A contains all elements that are not members of A. This is usually notated as A^c.

Taking the complement is an involution: (A^c)^c = A.

If A is open, then A^C is closed. If A is closed, then A^C is open.

Operations

The union of two sets contains all members of both. A ⋃ B = {x | x ∈ A or x ∈ B}.

The union of all subsets A_i can be expressed as:

The intersection of two sets contains all members that are common between the two. A ⋂ B = {x | x ∈ A and x ∈ B}.

The intersection of all subsets A_i can be expressed as:

A pair of sets are disjoint if there is no intersection, i.e., A ⋂ B = ∅.

The set difference of A with respect to B contains all members in A that are not in B. A \ B = {x ∈ A | x ∉ B}.

The Cartesian product of A and B is the set of all pairs (x,y). A × B = {(x,y) | x ∈ A and y ∈ B}.

Properties

Let U be the universe of set A. It is always true that A ⋂ A^c = ∅ and A ⋃ A^c = U. Furthermore, ∅^c = U and U^c = ∅.

De Morgan's laws prove that:

• (A ⋂ B)^c = A^c ⋃ B^c

• (A ⋃ B)^c = A^c ⋂ B^c

For set differences, ∅ has an identity property. A \ ∅ = A.

Lastly, note that A^c \ B^c = B \ A.

CategoryRicottone