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.

The empty set is notated Ø.

Commonly Used Sets

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.

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}.

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.

Cardinality

Cardinality is an extension of the concept of size, such that it is applicable to infinite sets.

∅ is considered to have cardinality of 0.

The countable case is intuitive. Consider the finite set A which contains natural numbers up to n (i.e., {1,2,...,n}. The cardinality of A is n: |A| = n.

This is generalized to infinite sets by functions. Given two sets A and B:

• If an injective function exists such that f : A -> B, then |A| ≤ |B|.

• If a bijective function exists such that f : A -> B, then |A| = |B|.

  • More formally: if injective functions f : A -> B and g : B -> A exist, then |A| ≤ |B| and |B| ≤ |A|, and finally |A| = |B|. This is the Cantor-Schröder-Bernstein theorem.

• If |A| ≤ |B| but |A| != |B|, then clearly |A| < |B|.

If |A| = |N| (i.e., A can be mapped to the natural numbers), then A is countably infinite. Otherwise an infinite set is uncountable.

Note that the set of all integers (Z) and the set of all rational numbers (Q) can be proven to be countably infinite.

Properties

Cardinality is...

• commutative: |A| = |B| -> |B| = |A|

• transitive: |A| = |B| and |B| = |C| -> |A| = |C|

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