Complement (set theory)

  (Redirected from Set-theoretic complement)

In set theory, the complement of a set A, often denoted by Ac (or A),[1] are the elements not in A.[2]

A circle filled with red inside a square. The area outside the circle is unfilled. The borders of both the circle and the square are black.
If A is the area colored red in this image…
An unfilled circle inside a square. The area inside the square not covered by the circle is filled with red. The borders of both the circle and the square are black.
… then the complement of A is everything else.

When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U that are not in A.

The relative complement of A with respect to a set B, also termed the set difference of B and A, written is the set of elements in B that are not in A.

Absolute complementEdit

 
The absolute complement of the white disc is the red region

DefinitionEdit

If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the relative complement of A in U:[3]

 

Or formally:

 

The absolute complement of A is usually denoted by Ac. Other notations include  [2]  [4]

ExamplesEdit

  • Assume that the universe is the set of integers. If A is the set of odd numbers, then the complement of A is the set of even numbers. If B is the set of multiples of 3, then the complement of B is the set of numbers congruent to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).
  • Assume that the universe is the standard 52-card deck. If the set A is the suit of spades, then the complement of A is the union of the suits of clubs, diamonds, and hearts. If the set B is the union of the suits of clubs and diamonds, then the complement of B is the union of the suits of hearts and spades.

PropertiesEdit

Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements:

De Morgan's laws:[5]

  •  
  •  

Complement laws:[5]

  •  
  •  
  •  
  •  
  •  
    (this follows from the equivalence of a conditional with its contrapositive).

Involution or double complement law:

  •  

Relationships between relative and absolute complements:

  •  
  •  

Relationship with a set difference:

  •  

The first two complement laws above show that if A is a non-empty, proper subset of U, then {A, Ac} is a partition of U.

Relative complementEdit

DefinitionEdit

If A and B are sets, then the relative complement of A in B,[5] also termed the set difference of B and A,[6] is the set of elements in B but not in A.

 
The relative complement of A (left circle) in B (right circle):  

The relative complement of A in B is denoted   according to the ISO 31-11 standard. It is sometimes written   but this notation is ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements   where b is taken from B and a from A.

Formally:

 

ExamplesEdit

  •  
  •  
  • If   is the set of real numbers and   is the set of rational numbers, then   is the set of irrational numbers.

PropertiesEdit

Let A, B, and C be three sets. The following identities capture notable properties of relative complements:

  •  
  •  
  •  
    with the important special case   demonstrating that intersection can be expressed using only the relative complement operation.
  •  
  •  
  •  
  •  
  •  
  •  

Complementary relationEdit

A binary relation   is defined as a subset of a product of sets   The complementary relation   is the set complement of   in   The complement of relation   can be written

 
Here,   is often viewed as a logical matrix with rows representing the elements of   and columns elements of   The truth of   corresponds to 1 in row   column   Producing the complementary relation to   then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.

Together with composition of relations and converse relations, complementary relations and the algebra of sets are the elementary operations of the calculus of relations.

LaTeX notationEdit

In the LaTeX typesetting language, the command \setminus[7] is usually used for rendering a set difference symbol, which is similar to a backslash symbol. When rendered, the \setminus command looks identical to \backslash, except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash}. A variant \smallsetminus is available in the amssymb package.

In programming languagesEdit

Some programming languages have sets among their builtin data structures. Such a data structure behaves as a finite set, that is, it consists of a finite number of data that are not specifically ordered, and may thus be considered as the elements of a set. In some cases, the elements are not necessary distinct, and the data structure codes multisets rather than sets. These programming languages have operators or functions for computing the complement and the set differences.

These operators may generally be applied also to data structures that are not really mathematical sets, such as ordered lists or arrays. It follows that some programming languages may have a function called set_difference, even if they do not have any data structure for sets.

See alsoEdit

NotesEdit

  1. ^ "Complement and Set Difference". web.mnstate.edu. Retrieved 2020-09-04.
  2. ^ a b "Complement (set) Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2020-09-04.
  3. ^ The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.
  4. ^ Bourbaki 1970, p. E II.6.
  5. ^ a b c Halmos 1960, p. 17.
  6. ^ Devlin 1979, p. 6.
  7. ^ [1] The Comprehensive LaTeX Symbol List

ReferencesEdit

External linksEdit