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Encyclopedia > Algebra of sets

The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality (mathematics) and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... 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. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In mathematics, the concept of binary relation is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... See also the disambiguation page title equality. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

Contents

Introduction

The algebra of sets is the development of the fundamental properties of set operations and set relations. These properties provide insight into the fundamental nature of sets. They also have practical considerations.


Just like expressions and calculations in ordinary arithmetic, expressions and calculations involving sets can be quite complex. It is helpful to have systematic procedures available for manipulating and evaluating such expressions and performing such computations.


In the case of arithmetic, it is elementary algebra that develops the fundamental properties of arithmetic operations and relations. Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. ...


For example, the operations of addition and multiplication obey familiar laws such as associativity, commutativity and distributivity, while, the "less than or equal" relation satisfies such laws as reflexivity, antisymmetry and transitivity. These laws provide tools which facilitate computation, as well as describe the fundamental nature of numbers, their operations and relations. Addition is one of the basic operations of arithmetic. ... In its simplest form, multiplication is a quick way of adding identical numbers. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, a binary relation R over a set X is reflexive if for all a in X, a is related to itself. ... In mathematics, a binary relation R on a set X is antisymmetric if it holds for all a and b in X that if a is related to b and b is related to a then a = b. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ...


The algebra of sets is the set-theoretic analogue of the algebra of numbers. It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. These are the topics covered in this article. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory (preferably after reading the other articles first). The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... Naive set theory1 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 certain axioms. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ...


The fundamental laws of set algebra

The binary operations of set union and intersection satisfy many identities. Several of these identities or "laws" have well established names. Three pairs of laws, are stated, without proof, in the following proposition. In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific 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. ... In mathematics, an identity is an equality that remains true regardless of the values of any variables that appear within it. ... In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ...


PROPOSITION 1: For any sets A, B, and C, the following identities hold: The notion of a set is one of the most important and fundamental concepts in modern mathematics. ...

commutative laws:
  • AB  =  BA
  • AB  =  BA
associative laws:
  • (AB) ∪C  =  A ∪(BC)
  • (AB) ∩C  =  A ∩(BC)
distributive laws:
  • A ∪(BC)  =  (AB) ∩(AC)
  • A ∩(BC)  =  (AB) ∪(AC)

Notice that the analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection distributes over unions. However, unlike addition and multiplication, union also distributes over intersection. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...


The next proposition, states two additional pairs of laws involving three specials sets: the empty set, the universal set and the complement of a set. In mathematics, the empty set is the set with no elements. ... 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. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...


PROPOSITION 2: For any subset A of universal set U, the following identities hold: A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

identity laws:
  • A ∪Ø  =  A
  • AU  =  A
complement laws:
  • AAC  =  U
  • AAC  =  Ø

The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity elements for union and intersection, respectively. In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...


Unlike addition and multiplication, union and intersection do not have inverse elements. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation of set complementation. In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In mathematics, a unary operation is an operation with only one operand. ...


The preceding five pairs of laws: the commutative, associative, distributive, identity and complement laws, can be said to encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.


The principle of duality

The above propositions display the following interesting pattern. Each of the identities stated above is one of a pair of identities, such that, each can be transformed into the other by interchanging ∪ and ∩, and also Ø and U.


These are examples of an extremely important and powerful property of set algebra, namely, the principle of duality for sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions and intersections, interchanging U and Ø and reversing inclusions is also true. A statement is said to be self-dual if it is equal to its own dual.


Some additional laws for unions and intersections

The following proposition states six more important laws of set algebra, involving unions and intersections.


PROPOSITION 3: For any subsets A and B of a universal set U, the following identities hold:

idempotent laws:
  • AA  =  A
  • AA  =  A
domination laws:
  • AU  =  U
  • A ∩Ø  =  Ø
absorption laws:
  • A ∪(AB)  =  A
  • A ∩(AB)  =  A

As noted above each of the laws stated in proposition 3, can be derived from the five fundamental pairs of laws stated in proposition 1 and proposition 2. As an illustration, a proof is given below for the idempotent law for union. In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...


Proof:

     AA = (AA) ∩U    by the identity law for intersection
  = (AA) ∩(AAC)    by the complement law for union
  = A ∪(AAC)    by the distributive law of union over intersection
  = A ∪Ø    by the complement law for intersection
  = A    by the identity law for union

The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law for union, namely the idempotent law for intersection.


Proof:

     AA = (AA) ∪Ø    by the identity law for union
  = (AA) ∪(AAC)    by the complement law for intersection
  = A ∩(AAC)    by the distributive law of intersection over union
  = AU    by the complement law for union
  = A    by the identity law for intersection

Some additional laws for complements

The following proposition states five more important laws of set algebra, involving complements.


PROPOSITION 4: Let A and B be subsets of a universe U, then:

De Morgan's laws:
  • (AB)C  =  ACBC
  • (AB)C  =  ACBC
double complement or Involution law:
  • ACC  =  A
complement laws for the universal set and the empty set:
  • ØC  =  U
  • UC  =  Ø

Notice that the double complement law is self-dual. In logic, De Morgans laws (or De Morgans theorem) are the two rules of propositional logic, boolean algebra and set theory not (P and Q) = (not P) or (not Q) not (P or Q) = (not P) and (not Q) which allow us to move a negation over a... This page is about involution in mathematics; for involution in philosophy and integral theory, see Involution (philosophy). ...


The next proposition, which is also self-dual, says that the complement of a set is the only set that satisfies the complement laws. In other words, complementation is characterized by the complement laws.


PROPOSITION 5: Let A and B be subsets of a universe U, then:

uniqueness of complements:
  • If AB  =  U, and AB  =  Ø then B = AC.

The algebra of inclusion

The following proposition says that inclusion is a partial order. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...


PROPOSITION 6: If A, B and C are sets then the following hold:

reflexivity:
  • A ⊆ A
antisymmetry:
  • A ⊆ B and B ⊆ A if and only if A = B
transitivity:
  • If A ⊆ B and B ⊆ C then A ⊆ C

The following proposition says that for any set S the power set of S ordered by inclusion is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra. In mathematics, a binary relation R over a set X is reflexive if for all a in X, a is related to itself. ... In mathematics, a binary relation R on a set X is antisymmetric if it holds for all a and b in X that if a is related to b and b is related to a then a = b. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which capture the essence of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and complement. ...


PROPOSITION 7: If A, B and C are subsets of a set S then the following hold:

existence of a least element and a greatest element:
  • Ø ⊆ A ⊆ S
existence of joins:
  • A ⊆ AB
  • If A ⊆ C and B ⊆ C then AB ⊆ C
existence of meets:
  • AB ⊆ A
  • If C ⊆ A and C ⊆ B then C ⊆ AB

The following proposition says that, the statement "A ⊆ B ", is equivalent to various other statements involving unions, intersections and complements. In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ...


PROPOSITION 8: For any two sets A and B, the following are equivalent:

  • A ⊆ B
  • AB  =  A
  • AB  =  B
  • A − B  =  Ø
  • BC ⊆ AC

The above proposition shows that the relation of set inclusion can be characterized by either of the operations of set union or set intersection, which means that the notion of set inclusion is axiomatically superfluous.


The algebra of relative complements

The following proposition, lists several identities concerning relative complements or set-theoretic difference. In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...


PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold:

  • C − (AB)  =  (C − A) ∪(CB)
  • C − (AB)  =  (C − A) ∩(CB)
  • C − (B − A)  =  (AC) ∪(C − B)
  • (B − A) ∩C  =  (BC) − A  =  B ∩(C − A)
  • (B − A) ∪C  =  (BC) − (A − C)
  • A − A  =  Ø
  • Ø − A  =  Ø
  • A − Ø  =  A
  • B − A  =  ACB
  • (B − A)C  =  ABC
  • U − A  =  AC
  • A − U  =  Ø

Further reading


  Results from FactBites:
 
The algebra of sets - Wikipedia, the free encyclopedia (1016 words)
The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion.
The algebra of sets is the development of the fundamental properties of set operations and set relations.
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion.
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Advanced algebra students should complete both this "Algebra by Chapter Series" and the "Advanced Math Series" which would then be the equivalent of 2 years of Algebra (Algebra 1 and Algebra 2) including a complete course of Trigonometry.
Learn algebra as each topic is introduced and explained with sample algebra problems followed by practice sets of algebra problems with the algebra solutions offered to the student step-by-step.
Algebra 1 students will learn this material for the first time and algebra 2 students will review this very important material which is necessary for more advanced algebra studies.
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