Complement (set theory)

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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, but not in A. Complement (set theory)_sentence_0

The relative complement of A with respect to a set B, also termed the set difference of B and A, written B \ A, is the set of elements in B but not in A. Complement (set theory)_sentence_1

Absolute complement Complement (set theory)_section_0

Definition Complement (set theory)_section_1

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). Complement (set theory)_sentence_2

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: Complement (set theory)_sentence_3

Or formally: Complement (set theory)_sentence_4

Examples Complement (set theory)_section_2

Complement (set theory)_unordered_list_0

  • 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).Complement (set theory)_item_0_0
  • 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.Complement (set theory)_item_0_1

Properties Complement (set theory)_section_3

Let A and B be two sets in a universe U. Complement (set theory)_sentence_5

The following identities capture important properties of absolute complements: Complement (set theory)_sentence_6

De Morgan's laws: Complement (set theory)_sentence_7

Complement laws: Complement (set theory)_sentence_8

Involution or double complement law: Complement (set theory)_sentence_9

Relationships between relative and absolute complements: Complement (set theory)_sentence_10

Relationship with set difference: Complement (set theory)_sentence_11

The first two complement laws above show that if A is a non-empty, proper subset of U, then {A, A} is a partition of U. Complement (set theory)_sentence_12

Relative complement Complement (set theory)_section_4

Definition Complement (set theory)_section_5

If A and B are sets, then the relative complement of A in B, also termed the set difference of B and A, is the set of elements in B but not in A. Complement (set theory)_sentence_13

The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard. Complement (set theory)_sentence_14

It is sometimes written B − A, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all elements b − a, where b is taken from B and a from A. Complement (set theory)_sentence_15

Formally: Complement (set theory)_sentence_16

Examples Complement (set theory)_section_6

Properties Complement (set theory)_section_7

See also: List of set identities and relations Complement (set theory)_sentence_17

Let A, B, and C be three sets. Complement (set theory)_sentence_18

The following identities capture notable properties of relative complements: Complement (set theory)_sentence_19

Complementary relation Complement (set theory)_section_8

Here, R is often viewed as a logical matrix with rows representing the elements of X, and columns elements of Y. Complement (set theory)_sentence_20

The truth of aRb corresponds to 1 in row a, column b. Complement (set theory)_sentence_21

Producing the complementary relation to R then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement. Complement (set theory)_sentence_22

Together with composition of relations and converse relations, complementary relations and the algebra of sets are the elementary operations of the calculus of relations. Complement (set theory)_sentence_23

LaTeX notation Complement (set theory)_section_9

In the LaTeX typesetting language, the command \setminus is usually used for rendering a set difference symbol, which is similar to a backslash symbol. Complement (set theory)_sentence_24

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}. Complement (set theory)_sentence_25

A variant \smallsetminus is available in the amssymb package. Complement (set theory)_sentence_26

In programming languages Complement (set theory)_section_10

Some programming languages have sets among their builtin data structures. Complement (set theory)_sentence_27

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. Complement (set theory)_sentence_28

In some cases, the elements are not necessary distinct, and the data structure codes multisets rather than sets. Complement (set theory)_sentence_29

These programming languages have operators or functions for computing the complement and the set differences. Complement (set theory)_sentence_30

These operators may generally be applied also to data structures that are not really mathematical sets, such as ordered lists or arrays. Complement (set theory)_sentence_31

It follows that some programming languages may have a function called set_difference, even if they do not have any data structure for sets. Complement (set theory)_sentence_32

See also Complement (set theory)_section_11

Complement (set theory)_unordered_list_1

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: (set theory).