Skip to content

Sets


A set contains elements. Conversely, an element is a member of a set. The following phrases are all used synonymously:
  • x is a member of y
  • x is contained in y
  • x is included in y
  • y contains x
  • y includes x
The curly brace notation { } is used to denote members of a set.  As an example, {1, 2, 3} denotes a set containing 1, 2, and 3 as members.


Empty or Null Set:


The set with no members is the empty or null set. The expressions

{}

{x : xx}

all specify the empty set.



Special Sets:

Some sets are used frequently in mathematics so they are given by special notation.

N = { Set of whole numbers} = {0, 1, 2, 3, 4, 5, ...}

Z = { Set of all integers} = {..., 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, ...}

N = { Set of rational numbers} = {a/b where a and b are integers and b ≠ 0}

= { Set of real numbers}


Subsets:

A set S is a subset of set A if every member of S is a member of A. We use the horseshoe notation to indicate subsets. The expression

{1, 2, 3} Í  {1, 2, 3, 4}

says that {1, 2, 3} is a subset of {1, 2, 3, 4}. The empty set is a subset of every set. Every set is a subset of itself. A proper subset A is a subset of A that is not identical with A. The expression of

{1, 2, 3}
Ì {1, 2, 3, 4}

says that {1, 2, 3} is a proper subset of {1, 2, 3, 4}.


Union:

Sets can be "added" together. The union of two sets A and B, denoted by A ∪ B, is the set of all things which are members of either A or B.



Some basic properties of unions are:
  • A ∪ B   =   B ∪ A
  • A  Í  A ∪ B
  • A ∪ A   =  A
  • A ∪ ∅   =  A
  • A  Í  B if and only if A ∪ B = B

Intersection:

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B  =  ∅, then A and B are said to be disjoint.


Some basic properties of intersections:
    • A ∩ B   =   B ∩ A
    • A ∩ B  Í  A
    • A ∩ A   =   A
    • A ∩ ∅   =   ∅
    • A  Í  B if and only if A ∩ B = A
  • Complements:

    Sets may be "subtracted" from one another. The relative complement of A in B, denoted by B \ A, (or B − A) is the set of all elements which are members of B, but not members of A.

    In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A, is called the absolute complement of A, and is denoted by A′.


    Some basic properties of complements:

    • A ∪ A′ = U
    • A ∩ A′ = ∅
    • (A′ )′ = A
    • A \ A = ∅
    • A \ B = A ∩ B′