absolute complement

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In
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written $B \setminus A,$ is the set of elements in that are not in .

# Absolute complement

## Definition

If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : $A^\complement = U \setminus A.$ Or formally: $A^\complement = \.$ The absolute complement of is usually denoted by Other notations include $\overline A, A\text{'},$ $\complement_U A, \text \complement A.$.

## Examples

* Assume that the universe is the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. If is the set of odd numbers, then the complement of is the set of even numbers. If is the set of multiples of 3, then the complement of 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
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. If the set is the suit of spades, then the complement of is the union of the suits of clubs, diamonds, and hearts. If the set is the union of the suits of clubs and diamonds, then the complement of is the union of the suits of hearts and spades.

## Properties

Let and be two sets in a universe . The following identities capture important properties of absolute complements:
De Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathe ...
: * $\left\left(A \cup B \right\right)^\complement = A^\complement \cap B^\complement.$ * $\left\left(A \cap B \right\right)^\complement = A^\complement \cup B^\complement.$ Complement laws: * $A \cup A^\complement = U.$ * $A \cap A^\complement = \varnothing .$ * $\varnothing^\complement = U.$ * $U^\complement = \varnothing.$ * $\textA\subseteq B\textB^\complement \subseteq A^\complement.$ *: (this follows from the equivalence of a conditional with its contrapositive). Involution or double complement law: * $\left\left(A^\complement\right\right)^\complement = A.$ Relationships between relative and absolute complements: * $A \setminus B = A \cap B^\complement.$ * $\left(A \setminus B\right)^\complement = A^\complement \cup B = A^\complement \cup \left(B \cap A\right).$ Relationship with a set difference: * $A^\complement \setminus B^\complement = B \setminus A.$ The first two complement laws above show that if is a non-empty, proper subset of , then is a partition of .

# Relative complement

## Definition

If and are sets, then the relative complement of in ,. also termed the set difference of and ,. is the set of elements in but not in . The relative complement of in is denoted $B \setminus A$ according to the ISO 31-11 standard. It is sometimes written $B - A,$ but this notation is ambiguous, as in some contexts (for example, Minkowski set operations in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
) it can be interpreted as the set of all elements $b - a,$ where is taken from and from . Formally: $B \setminus A = \.$

## Examples

* $\ \setminus \ = \.$ * $\ \setminus \ = \ .$ * If $\mathbb$ is the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s and $\mathbb$ is the set of rational numbers, then $\mathbb\setminus\mathbb$ is the set of
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two i ...
s.

## Properties

Let , , and be three sets. The following identities capture notable properties of relative complements: :* $C \setminus \left(A \cap B\right) = \left(C \setminus A\right) \cup \left(C \setminus B\right).$ :* $C \setminus \left(A \cup B\right) = \left(C \setminus A\right) \cap \left(C \setminus B\right).$ :* $C \setminus \left(B \setminus A\right) = \left(C \cap A\right) \cup \left(C \setminus B\right),$ :*:with the important special case $C \setminus \left(C \setminus A\right) = \left(C \cap A\right)$ demonstrating that intersection can be expressed using only the relative complement operation. :* $\left(B \setminus A\right) \cap C = \left(B \cap C\right) \setminus A = B \cap \left(C \setminus A\right).$ :* $\left(B \setminus A\right) \cup C = \left(B \cup C\right) \setminus \left(A \setminus C\right).$ :* $A \setminus A = \empty.$ :* $\empty \setminus A = \empty.$ :* $A \setminus \empty = A.$ :* $A \setminus U = \empty.$ :* If $A\subset B$, then $C\setminus A\supset C\setminus B$. :* $A \supseteq B \setminus C$ is equivalent to $C \supseteq B \setminus A$.

# Complementary relation

A binary relation $R$ is defined as a subset of a product of sets $X \times Y.$ The complementary relation $\bar$ is the set complement of $R$ in $X \times Y.$ The complement of relation $R$ can be written $\bar \ = \ (X \times Y) \setminus R.$ Here, $R$ is often viewed as a logical matrix with rows representing the elements of $X,$ and columns elements of $Y.$ The truth of $aRb$ corresponds to 1 in row $a,$ column $b.$ 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. Together with
composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
and converse relations, complementary relations and the
algebra of sets In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the re ...
are the elementary
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s of the
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.

# LaTeX notation

In the LaTeX typesetting language, the command \setminus
The Comprehensive LaTeX Symbol List
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. A variant \smallsetminus is available in the amssymb package. The symbol $\complement$ (as opposed to $C$) is produced by \complement. (It corresponds to the Unicode symbol ∁.)

# In programming languages

Some programming languages have sets among their builtin data structures. Such a data structure behaves as a
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. ...
, 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
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...
s 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
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. It follows that some programming languages may have a function called set_difference, even if they do not have any data structure for sets.