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Encyclopedia > Complement (set theory)

In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

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Relative complement

If A and B are sets, then the relative complement of A in B, also known as the set-theoretic difference of B and A, is the set of elements in B, but not in A. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

The relative complement of A in B

The relative complement of A in B is denoted B  A (sometimes written B − A, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all b − a, where b is taken from B and a from A). Image File history File links B_minus_A,_relative_complement_(set_teory,_Venn_diagram). ... Image File history File links B_minus_A,_relative_complement_(set_teory,_Venn_diagram). ...


Formally:

B setminus A = { xin B , | , x notin A }.

Examples:

The following proposition lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...


PROPOSITION 1: If A, B, and C are sets, then the following identities hold: In mathematics, an identity is an equality that remains true regardless of the values of any variables that appear within it. ...

  • C  (A ∩ B)  =  (C  A)∪(C  B)
  • C  (A ∪ B)  =  (C  A)∩(C  B)
  • C  (B  A)  =  (A ∩ C)∪(C  B)
  • (B  A) ∩ C  =  (B ∩ C)  A  =  B∩(C  A)
  • (B  A) ∪ C  =  (B ∪ C)  (A  C)
  • A  A  =  Ø
  • Ø  A  =  Ø
  • A  Ø  =  A

Practical details

In the LaTeX typesetting language the command setminus is usually used for rendering a set difference symbol – a backslash-like 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 ,backslash, . This article is about the typesetting system. ... The backslash ( ) is a typographical mark (glyph) used chiefly in computing. ...


The Mathematica programming language implements the operation with the Complement function. For other uses, see Mathematica (disambiguation). ...


The Matlab programming language implements the operation with the setdiff function. Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ...


Absolute complement

The complement of A in U
The complement of A in U

If a universe U is defined, then the relative complement of A in U is called the absolute complement (or simply complement) of A, and is denoted by AC (or sometimes A′, also the same set often is denoted by complement_U A or complement A if U is fixed), that is: Image File history File links Absolute_complement. ... Image File history File links Absolute_complement. ... In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ...

AC  = U  A.

For example, if the universe is the set of natural numbers, then the complement of the set of odd numbers is the set of even numbers. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...


The following proposition lists some important properties of absolute complements in relation to the set-theoretic operations of union and intersection. In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...


PROPOSITION 2: If A and B are subsets of a universe U, then the following identities hold: In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ...

De Morgan's laws:
  • (A ∪ B)C  = AC ∩ BC
  • (A ∩ B)C  = AC ∪ BC
Complement laws:
  • A ∪ AC  =  U
  • A ∩ AC  =  Ø
  • ØC  =  U
  • UC  =  Ø
  • If AB, then BCAC (this follows from the equivalence of a conditional with its contrapositive)
Involution or double complement law:
  • ACC  =  A.
Relationships between relative and absolute complements:
  • A  B = A ∩ BC
  • (A  B)C = AC ∪ B

The first two complement laws above shows that if A is a non-empty subset of U, then {A, AC} is a partition of U. note that demorgans laws are also a big part in circut design. ... In predicate logic, the contrapositive (or transposition) of the statement p implies q is not-q implies not-p. ... In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ... A partition of U into 6 blocks: an Euler diagram representation. ...


See also


  Results from FactBites:
 
Set theory Summary (2598 words)
In naive set theory, sets are introduced and understood using what is taken to be the self-evident concept of sets as collections of objects considered as a whole.
Naive set theory is the original set theory developed by mathematicians at the end of the 19th century.
Axiomatic set theory is a rigorous axiomatic branch of mathematics developed in response to the discovery of serious flaws (such as Russell's paradox) in naïve set theory.
Naive set theory (1783 words)
Naive set theory is distinguished from axiomatic set theory by the fact that the former regards sets as collections of objects, called the elements or members of the set, whereas the latter regards sets only as that which satisfies the axioms.
Sets are of great importance in mathematics; in fact, in the modern formal treatment, the whole machinery of pure mathematics (numbers, relations, functions, etc.) is defined in terms of sets.
Naive set theory was developed at the end of the 19th century (principally by Georg Cantor) in order to allow mathematicians to work with infinite sets consistently.
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