Intersection (set theory)From Wikipedia, the free encyclopedia

Contents

1 Algebra of sets 11.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The fundamental laws of set algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The principle of duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Some additional laws for unions and intersections . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Some additional laws for complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 The algebra of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 The algebra of relative complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Binary relation 62.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Disjoint sets 163.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Disjoint unions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Disjoint union 204.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Set theory denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Category theory point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Empty set 225.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Family of sets 286.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Intersection (set theory) 307.1 Basic denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.1.1 Intersecting and disjoint sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.2 Arbitrary intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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7.3 Nullary intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 Subset 368.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9 Union (set theory) 409.1 Union of two sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.2 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.3 Finite unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4 Arbitrary unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.4.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4.2 Union and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Chapter 1

Algebra of sets

The algebra of sets denes the properties and laws of sets, the set-theoretic operations of union, intersection, andcomplementation and the relations of set equality and set inclusion. It also provides systematic procedures for evalu-ating expressions, and performing calculations, involving these operations and relations.Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union,the meet operator being intersection, and the complement operator being set complement.

1.1 FundamentalsThe algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition andmultiplicationare associative and commutative, so are set union and intersection; just as the arithmetic relation less than or equalis reexive, antisymmetric and transitive, so is the set relation of subset.It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations ofequality and inclusion. 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.

1.2 The fundamental laws of set algebraThe binary operations of set union ( [ ) and intersection ( \ ) satisfy many identities. Several of these identities orlaws have well established names.

Commutative laws:

A [B = B [A A \B = B \A

Associative laws:

(A [B) [ C = A [ (B [ C) (A \B) \ C = A \ (B \ C)

Distributive laws:

A [ (B \ C) = (A [B) \ (A [ C) A \ (B [ C) = (A \B) [ (A \ C)

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 inter-section distributes over unions. However, unlike addition and multiplication, union also distributes over intersection.

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2 CHAPTER 1. ALGEBRA OF SETS

Two additional pairs of laws involve the special sets called the empty set and the universal set U ; together withthe complement operator (AC denotes the complement of A). The empty set has no members, and the universal sethas all possible members (in a particular context).

Identity laws:

A [? = A A \ U = A

Complement laws:

A [AC = U A \AC = ?

The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, and U are the identity elements for union and intersection, respectively.Unlike addition and multiplication, union and intersection do not have inverse elements. However the complementlaws give the fundamental properties of the somewhat inverse-like unary operation of set complementation.The preceding ve pairs of lawsthe commutative, associative, distributive, identity and complement lawsencompassall of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.Note that if the complement laws are weakened to the rule (AC)C = A , then this is exactly the algebra of proposi-tional linear logic.

1.3 The principle of dualitySee also: Duality (order theory)

Each of the identities stated above is one of a pair of identities such that each can be transformed into the other byinterchanging and , and also and U.These are examples of an extremely important and powerful property of set algebra, namely, the principle of dualityfor sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions andintersections, interchanging U and and reversing inclusions is also true. A statement is said to be self-dual if it isequal to its own dual.

1.4 Some additional laws for unions and intersectionsThe 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:

A [A = A A \A = A

domination laws:

A [ U = U A \? = ?

absorption laws:

A [ (A \B) = A A \ (A [B) = A

1.5. SOME ADDITIONAL LAWS FOR COMPLEMENTS 3

As noted above, each of the laws stated in proposition 3 can be derived from the ve fundamental pairs of laws statedabove. As an illustration, a proof is given below for the idempotent law for union.Proof:The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law forunion, namely the idempotent law for intersection.Proof:Intersection can be expressed in terms of set dierence :A \B = Ar (ArB)

1.5 Some additional laws for complementsThe following proposition states ve more important laws of set algebra, involving complements.PROPOSITION 4: Let A and B be subsets of a universe U, then:

De Morgans laws:

(A [B)C = AC \BC (A \B)C = AC [BC

double complement or Involution law:

(AC)C = Acomplement laws for the universal set and the empty set:

?C = U UC = ?

Notice that the double complement law is self-dual.The next proposition, which is also self-dual, says that the complement of a set is the only set that satises thecomplement 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 A [B = U , and A \B = ? , then B = AC

1.6 The algebra of inclusionThe following proposition says that inclusion, that is the binary relation of one set being a subset of another, is apartial order.PROPOSITION 6: If A, B and C are sets then the following hold:

reexivity:

A A

antisymmetry:

A B and B A if and only if A = B

transitivity:

4 CHAPTER 1. ALGEBRA OF SETS

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, andhence together with the distributive and complement laws above, show that it is a Boolean algebra.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: ? A S

existence of joins: A A [B If A C and B C , then A [B C

existence of meets: A \B A If C A and C B , then C A \B

The following proposition says that the statement A B is equivalent to various other statements involving unions,intersections and complements.PROPOSITION 8: For any two sets A and B, the following are equivalent:

A B A \B = A A [B = B ArB = ? BC AC

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

1.7 The algebra of relative complementsThe following proposition lists several identities concerning relative complements and set-theoretic dierences.PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold:

C n (A \B) = (C nA) [ (C nB) C n (A [B) = (C nA) \ (C nB) C n (B nA) = (A \ C) [ (C nB) (B nA) \ C = (B \ C) nA = B \ (C nA) (B nA) [ C = (B [ C) n (A n C) A nA = ? ? nA = ? A n? = A B nA = AC \B (B nA)C = A [BC U nA = AC A n U = ?

1.8. SEE ALSO 5

1.8 See also -algebra is an algebra of sets, completed to include countably innite operations. Axiomatic set theory Field of sets Naive set theory Set (mathematics)

1.9 References Stoll, Robert R.; Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4. TheAlgebra of Sets, pp 1623

Courant, Richard, Herbert Robbins, Ian Stewart,What is mathematics?: An Elementary Approach to Ideas andMethods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. SUPPLEMENT TO CHAPTER IITHE ALGEBRA OF SETS

1.10 External links Operations on Sets at ProvenMath

Chapter 2

Binary relation

Relation (mathematics)" redirects here. For a more general notion of relation, see nitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A A. More generally, a binary relation between two sets A and B is a subsetof A B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include 4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", anddivides in arithmetic, "is congruent to" in geometry, is adjacent to in graph theory, is orthogonal to in linearalgebra and many more. The concept of function is dened as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R A1 An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZZZ is lies between ... and ..., containing e.g. the triples (5,2,8), (5,8,2), and (4,9,7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of is an element of or is a subset of in settheory, without running into logical inconsistencies such as Russells paradox.

2.1 Formal denition

A binary relation R is usually dened as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as dened by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X Y, and from X to Y" must always be either specied or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

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2.2. SPECIAL TYPES OF BINARY RELATIONS 7

2.1.1 Is a relation more than its graph?According to the denition above, two relations with identical graphs but dierent domains or dierent codomainsare considered dierent. For example, ifG = f(1; 2); (1; 3); (2; 7)g , then (Z;Z; G) , (R;N; G) , and (N;R; G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often dened as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then dened as the set of all x such that there exists at least oney such that (x; y) 2 R , the range of R is dened as the set of all y such that there exists at least one x such that(x; y) 2 R , and the eld of R is the union of its domain and its range.[2][3][4]A special case of this dierence in points of view applies to the notion of function. Many authors insist on distin-guishing between a functions codomain and its range. Thus, a single rule, like mapping every real number x tox2, can lead to distinct functions f : R ! R and f : R ! R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique rst components. This dierence in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivityor being ontoas a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the denitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodenitions usually matters only in very formal contexts, like category theory.

2.1.2 ExampleExample: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation is owned by is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the rst element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by RJ means that the ball is owned by John.Two dierent relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is dierent from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identied or even dened as G(R) and an ordered pair (x, y) G(R)" is usually denoted as"(x, y) R".

2.2 Special types of binary relationsSome important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be dierent sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = 5and z = +5 to y = 25.

functional (also called univalent[8] or right-unique[7] or right-denite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=5 and z=+5.

8 CHAPTER 2. BINARY RELATION

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is dierentfrom the denition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = 14 to any real number y.

surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = 14.

Uniqueness and totality properties:

2.3. RELATIONS OVER A SET 9

A function: a relation that is functional and left-total. Both the green and the red relation are functions. An injective function: a relation that is injective, functional, and left-total. A surjective function or surjection: a relation that is functional, left-total, and right-total. A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

2.2.1 DifunctionalLess commonly encountered is the notion of difunctional (or regular) relation, dened as a relation R such thatR=RR1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element xX to a set xR = { yY| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can dene the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R x2R implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A Cand g: B C and then dene the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) A B | f(a) = g(b) }. Every difunctional relation R A B arises as the joint kernel of two functionsf: A C and g: B C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justied by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term rectangular to denote any heterogeneous relation whatsoever.[6]

2.3 Relations over a setIf X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

reexive: for all x in X it holds that xRx. For example, greater than or equal to () is a reexive relation butgreater than (>) is not.

irreexive (or strict): for all x in X it holds that not xRx. For example, > is an irreexive relation, but is not. coreexive: for all x and y in X it holds that if xRy then x = y. An example of a coreexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reexive and coreexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreexive, nor coreexive, nor reexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

symmetric: for all x and y in X it holds that if xRy then yRx. Is a blood relative of is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.

10 CHAPTER 2. BINARY RELATION

antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, is anti-symmetric (so is >, butonly because the condition in the denition is always false).[18]

asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreexive.[19] For example, > is asymmetric, but is not.

transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreexive if andonly if it is asymmetric.[20] For example, is ancestor of is transitive, while is parent of is not.

total: for all x and y in X it holds that xRy or yRx (or both). This denition for total is dierent from left totalin the previous section. For example, is a total relation.

trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation divides on natural numbers is not.[21]

Right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz. Left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz. Euclidean: An Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because ifx=y and x=z, then y=z.

serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the denition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reexive. A relation that is only symmetric and transitive (without necessarily beingreexive) is called a partial equivalence relation.A relation that is reexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

2.4 Operations on binary relationsIf R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

Union: R S X Y, dened as R S = { (x, y) | (x, y) R or (x, y) S }. For example, is the union of >and =.

Intersection: R S X Y, dened as R S = { (x, y) | (x, y) R and (x, y) S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

Composition: S R, also denoted R ; S (or more ambiguously R S), dened as S R = { (x, z) | there existsy Y, such that (x, y) R and (y, z) S }. The order of R and S in the notation S R, used here agrees withthe standard notational order for composition of functions. For example, the composition is mother of isparent of yields is maternal grandparent of, while the composition is parent of is mother of yields isgrandmother of.

2.4. OPERATIONS ON BINARY RELATIONS 11

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin .If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

Inverse or converse: R 1, dened as R 1 = { (y, x) | (x, y) R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, is less than ().

If R is a binary relation over X, then each of the following is a binary relation over X:

Reexive closure: R =, dened as R = = { (x, x) | x X } R or the smallest reexive relation over X containingR. This can be proven to be equal to the intersection of all reexive relations containing R.

Reexive reduction: R , dened as R = R \ { (x, x) | x X } or the largest irreexive relation over Xcontained in R.

Transitive closure: R +, dened as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

Transitive reduction: R , dened as a minimal relation having the same transitive closure as R. Reexive transitive closure: R *, dened as R * = (R +) =, the smallest preorder containing R. Reexive transitive symmetric closure: R , dened as the smallest equivalence relation over X containingR.

2.4.1 ComplementIf R is a binary relation over X and Y, then the following too:

The complement S is dened as x S y if not x R y. For example, on real numbers, is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

If a relation is symmetric, the complement is too. The complement of a reexive relation is irreexive and vice versa. The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

2.4.2 RestrictionThe restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reexive, irreexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of is parent of is is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

12 CHAPTER 2. BINARY RELATION

Also, the various concepts of completeness (not to be confused with being total) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

2.4.3 Algebras, categories, and rewriting systemsVarious operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in nitary relations (and in practice also niteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

2.5 Sets versus classesCertain mathematical relations, such as equal to, member of, and subset of, cannot be understood to be binaryrelations as dened above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of equality as a binary relation =, wemust take the domain and codomain to be the class of all sets, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a large enough set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the subset of relation needs to be restricted to have domain and codomain P(A) (the power set ofa specic set A): the resulting set relation can be denoted A. Also, the member of relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation A that is a set. Bertrand Russell has shown thatassuming to be dened on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or MorseKelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modication needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this denition one can for instance dene a functionrelation between every set and its power set.

2.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

The number of irreexive relations is the same as that of reexive relations. The number of strict partial orders (irreexive transitive relations) is the same as that of partial orders. The number of strict weak orders is the same as that of total preorders. The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

the number of equivalence relations is the number of partitions, which is the Bell number.

2.7. EXAMPLES OF COMMON BINARY RELATIONS 13

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

2.7 Examples of common binary relations order relations, including strict orders:

greater than greater than or equal to less than less than or equal to divides (evenly) is a subset of

equivalence relations: equality is parallel to (for ane spaces) is in bijection with isomorphy

dependency relation, a nite, symmetric, reexive relation. independency relation, a symmetric, irreexive relation which is the complement of some dependency relation.

2.8 See also Conuence (term rewriting) Hasse diagram Incidence structure Logic of relatives Order theory Triadic relation

2.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 13301331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; PanosM. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science&BusinessMedia. pp. 299300. ISBN 978-0-387-74758-3.

14 CHAPTER 2. BINARY RELATION

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. xxi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four denitions appear in the following:

Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 1921. ISBN 978-0-13-460643-9.

Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 2122. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Ms, Stephan (2007), Reasoning on Spatial Semantic Integrity Constraints, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 1923, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of correspondence here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). Semantics of Fuzzy Sets in Rough Set Theory. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). Coalgebraic Simulations and Congruences. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Bchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 3537. ISBN 978-1-4613-8853-1.

[16] M. E. Mller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] Flaka, V.; Jeek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as strictlyantisymmetric.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). Generalization of rough sets using relationships between attribute values (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 3033..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

2.10 References M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

2.11. EXTERNAL LINKS 15

2.11 External links Hazewinkel, Michiel, ed. (2001), Binary relation, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 3

Disjoint sets

This article is about the mathematical concept. For the data structure, see Disjoint-set data structure.In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are

A BTwo disjoint sets.

sets whose intersection is the empty set.[1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and{3, 4, 5} are not.

3.1 Generalizations

This denition of disjoint sets can be extended to any family of sets. A family of sets is pairwise disjoint ormutuallydisjoint if every two dierent sets in the family are disjoint.[1] For example, the collection of sets { {1}, {2}, {3}, ...} is pairwise disjoint.Two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two innite setswhose intersection is a nite set may be said to be almost disjoint.[2]

In topology, there are various notions of separated sets with more strict conditions than disjointness. For instance,two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods. Similarly, ina metric space, positively separated sets are sets separated by a nonzero distance.[3]

16

3.2. EXAMPLES 17

A

BC

A pairwise disjoint family of sets

3.2 Examples

The set of the drum and the guitar is disjoint to the set of the card and the book

A pairwise disjoint family of sets

A non pairwise disjoint family of sets

3.3 IntersectionsDisjointness of two sets, or of a family of sets, may be expressed in terms of their intersections.

18 CHAPTER 3. DISJOINT SETS

Two sets A and B are disjoint if and only if their intersection A\B is the empty set.[1] It follows from this denitionthat every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself.[4]

A family F of sets is pairwise disjoint if, for every two sets in the family, their intersection is empty.[1] If the familycontains more than one set, this implies that the intersection of the whole family is also empty. However, a familyof only one set is pairwise disjoint, regardless of whether that set is empty, and may have a non-empty intersection.Additionally, a family of sets may have an empty intersection without being pairwise disjoint.[5] For instance, thethree sets { {1, 2}, {2, 3}, {1, 3} } have an empty intersection but are not pairwise disjoint. In fact, there are no twodisjoint sets in this collection. Also the empty family of sets is pairwise disjoint.[6]

A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that arepairwise disjoint. For instance, the closed intervals of the real numbers form a Helly family: if a family of closedintervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it mustbe pairwise disjoint.[7]

3.4 Disjoint unions and partitionsA partition of a set X is any collection of mutually disjoint non-empty sets whose union is X.[8] Every partition canequivalently be described by an equivalence relation, a binary relation that describes whether two elements belongto the same set in the partition.[8] Disjoint-set data structures[9] and partition renement[10] are two techniques incomputer science for eciently maintaining partitions of a set subject to, respectively, union operations that mergetwo sets or renement operations that split one set into two.A disjoint union may mean one of two things. Most simply, it may mean the union of sets that are disjoint.[11] Butif two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make themdisjoint before forming the union of the modied sets.[12] For instance two sets may be made disjoint by replacingeach element by an ordered pair of the element and a binary value indicating whether it belongs to the rst or secondset.[13] For families of more than two sets, one may similarly replace each element by an ordered pair of the elementand the index of the set that contains it.[14]

3.5 See also Hyperplane separation theorem for disjoint convex sets Mutually exclusive events Relatively prime, numbers with disjoint sets of prime divisors Set packing, the problem of nding the largest disjoint subfamily of a family of sets

3.6 References[1] Halmos, P. R. (1960), Naive Set Theory, Undergraduate Texts in Mathematics, Springer, p. 15, ISBN 9780387900926.

[2] Halbeisen, Lorenz J. (2011), Combinatorial Set Theory: With a Gentle Introduction to Forcing, Springer monographs inmathematics, Springer, p. 184, ISBN 9781447121732.

[3] Copson, Edward Thomas (1988),Metric Spaces, Cambridge Tracts in Mathematics 57, Cambridge University Press, p. 62,ISBN 9780521357326.

[4] Oberste-Vorth, Ralph W.; Mouzakitis, Aristides; Lawrence, Bonita A. (2012), Bridge to Abstract Mathematics, MAAtextbooks, Mathematical Association of America, p. 59, ISBN 9780883857793.

[5] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2010), A Transition to Advanced Mathematics, Cengage Learning,p. 95, ISBN 9780495562023.

[6] See answers to the question Is the empty family of sets pairwise disjoint?

[7] Bollobs, Bla (1986), Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability, Cam-bridge University Press, p. 82, ISBN 9780521337038.

3.7. EXTERNAL LINKS 19

[8] Halmos (1960), p. 28.

[9] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Cliord (2001), Chapter 21: Data structures forDisjoint Sets, Introduction to Algorithms (Second ed.), MIT Press, pp. 498524, ISBN 0-262-03293-7.

[10] Paige, Robert; Tarjan, Robert E. (1987), Three partition renement algorithms, SIAM Journal on Computing 16 (6):973989, doi:10.1137/0216062, MR 917035.

[11] Ferland, Kevin (2008), Discrete Mathematics: An Introduction to Proofs and Combinatorics, Cengage Learning, p. 45,ISBN 9780618415380.

[12] Arbib, Michael A.; Kfoury, A. J.; Moll, Robert N. (1981), A Basis for Theoretical Computer Science, The AKM series inTheoretical Computer Science: Texts and monographs in computer science, Springer-Verlag, p. 9, ISBN 9783540905738.

[13] Monin, Jean Franois; Hinchey,Michael Gerard (2003),Understanding FormalMethods, Springer, p. 21, ISBN9781852332471.

[14] Lee, John M. (2010), Introduction to Topological Manifolds, Graduate Texts in Mathematics 202 (2nd ed.), Springer, p.64, ISBN 9781441979407.

3.7 External links Weisstein, Eric W., Disjoint Sets, MathWorld.

Chapter 4

Disjoint union

In set theory, the disjoint union (or discriminated union) of a family of sets is a modied union operation thatindexes the elements according to which set they originated in. Or slightly dierent from this, the disjoint union ofa family of subsets is the usual union of the subsets which are pairwise disjoint disjoint sets means they have noelement in common.Note that these two concepts are dierent but strongly related. Moreover, it seems that they are essentially identicalto each other in category theory. That is, both are realizations of the coproduct of category of sets.

4.1 ExampleDisjoint union of sets A0 = {1, 2, 3} and A1 = {1, 2} can be computed by nding:

A0 = f(1; 0); (2; 0); (3; 0)gA1 = f(1; 1); (2; 1)gso

A0 tA1 = A0 [A1 = f(1; 0); (2; 0); (3; 0); (1; 1); (2; 1)g

4.2 Set theory denitionFormally, let {Ai : i I} be a family of sets indexed by i. The disjoint union of this family is the set

Gi2I

Ai =[i2If(x; i) : x 2 Aig:

The elements of the disjoint union are ordered pairs (x, i). Here i serves as an auxiliary index that indicates which Aithe element x came from.Each of the sets Ai is canonically isomorphic to the set

Ai = f(x; i) : x 2 Aig:Through this isomorphism, one may consider that Ai is canonically embedded in the disjoint union. For i j, the setsAi* and Aj* are disjoint even if the sets Ai and Aj are not.In the extreme case where each of the Ai is equal to some xed set A for each i I, the disjoint union is the Cartesianproduct of A and I:

20

4.3. CATEGORY THEORY POINT OF VIEW 21

Gi2I

Ai = A I:

One may occasionally see the notation

Xi2I

Ai

for the disjoint union of a family of sets, or the notation A + B for the disjoint union of two sets. This notation ismeant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the termsin the family. Compare this to the notation for the Cartesian product of a family of sets.Disjoint unions are also sometimes writtenUi2I Ai or Si2I Ai .In the language of category theory, the disjoint union is the coproduct in the category of sets. It therefore satises theassociated universal property. This also means that the disjoint union is the categorical dual of the Cartesian productconstruction. See coproduct for more details.For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying abuse of notation,the indexed family can be treated simply as a collection of sets. In this case Ai is referred to as a copy of Ai and thenotation S

A2CA is sometimes used.

4.3 Category theory point of viewIn category theory the disjoint union is dened as a coproduct in the category of sets.As such, the disjoint union is dened up to an isomorphism, and the above denition is just one realization of thecoproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct.This justies the second denition in the lead.This categorical aspect of the disjoint union explains why` is frequently used, instead ofF , to denote coproduct.4.4 See also

Coproduct Disjoint union (topology) Disjoint union of graphs Partition of a set Sum type Tagged union Union (computer science)

4.5 References Lang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised thirded.), New York: Springer-Verlag, p. 60, ISBN 978-0-387-95385-4

Weisstein, Eric W., Disjoint Union, MathWorld.

Chapter 5

Empty set

"" redirects here. For similar symbols, see (disambiguation).In mathematics, and more specically set theory, the empty set is the unique set having no elements; its size or

cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists byincluding an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of setsare trivially true for the empty set.Null set was once a common synonym for empty set, but is now a technical term in measure theory. The empty setmay also be called the void set.

5.1 NotationCommon notations for the empty set include "{}", "", and " ; ". The latter two symbols were introduced by theBourbaki group (specically Andr Weil) in 1939, inspired by the letter in the Norwegian and Danish alphabets(and not related in any way to the Greek letter ).[1]

The empty-set symbol is found at Unicode point U+2205.[2] In TeX, it is coded as \emptyset or \varnothing.

5.2 PropertiesIn standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements;therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of the emptyset rather than an empty set.The mathematical symbols employed below are explained here.For any set A:

The empty set is a subset of A:8A : ; A

The union of A with the empty set is A:8A : A [ ; = A

The intersection of A with the empty set is the empty set:8A : A \ ; = ;

The Cartesian product of A and the empty set is the empty set:8A : A ; = ;

22

5.2. PROPERTIES 23

The empty set is the set containing no elements.

The empty set has the following properties:

Its only subset is the empty set itself:8A : A ; ) A = ;

The power set of the empty set is the set containing only the empty set:2; = f;g

24 CHAPTER 5. EMPTY SET

A symbol for the empty set

Its number of elements (that is, its cardinality) is zero:card(;) = 0

The connection between the empty set and zero goes further, however: in the standard set-theoretic denition ofnatural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.For any property:

For every element of ; the property holds (vacuous truth); There is no element of ; for which the property holds.

Conversely, if for some property and some set V, the following two statements hold:

For every element of V the property holds; There is no element of V for which the property holds,

V = ;

5.3. IN OTHER AREAS OF MATHEMATICS 25

By the denition of subset, the empty set is a subset of any set A, as every element x of ; belongs to A. If it is nottrue that every element of ; is in A, there must be at least one element of ; that is not present in A. Since there areno elements of ; at all, there is no element of ; that is not in A. Hence every element of ; is in A, and ; is a subsetof A. Any statement that begins for every element of ; " is not making any substantive claim; it is a vacuous truth.This is often paraphrased as everything is true of the elements of the empty set.

5.2.1 Operations on the empty setOperations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sumof the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product).Ultimately, the results of these operations say more about the operation in question than about the empty set. Forinstance, zero is the identity element for addition, and one is the identity element for multiplication.A disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is adisarrangment of itself as no element can be found that retains its original position.

5.3 In other areas of mathematics

5.3.1 Extended real numbersSince the empty set has no members, when it is considered as a subset of any ordered set, then every member ofthat set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of thereal numbers, with its usual ordering, represented by the real number line, every real number is both an upper andlower bound for the empty set.[3] When considered as a subset of the extended reals formed by adding two numbersor points to the real numbers, namely negative innity, denoted 1; which is dened to be less than every otherextended real number, and positive innity, denoted +1; which is dened to be greater than every other extendedreal number, then:

sup ; = min(f1;+1g [ R) = 1;

and

inf ; = max(f1;+1g [ R) = +1:

That is, the least upper bound (sup or supremum) of the empty set is negative innity, while the greatest lower bound(inf or inmum) is positive innity. By analogy with the above, in the domain of the extended reals, negative innityis the identity element for the maximum and supremum operators, while positive innity is the identity element forminimum and inmum.

5.3.2 TopologyConsidered as a subset of the real number line (or more generally any topological space), the empty set is both closedand open; it is an example of a clopen set. All its boundary points (of which there are none) are in the empty set,and the set is therefore closed; while for every one of its points (of which there are again none), there is an openneighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the factthat every nite set is compact.The closure of the empty set is empty. This is known as preservation of nullary unions.

5.3.3 Category theoryIf A is a set, then there exists precisely one function f from {} to A, the empty function. As a result, the empty set isthe unique initial object of the category of sets and functions.

26 CHAPTER 5. EMPTY SET

The empty set can be turned into a topological space, called the empty space, in just one way: by dening the emptyset to be open. This empty topological space is the unique initial object in the category of topological spaces withcontinuous maps.The empty set is more ever a strict initial object: only the empty set has a function to the empty set.

5.4 Questioned existence

5.4.1 Axiomatic set theory

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness followsfrom the axiom of extensionality. However, the axiom of empty set can be shown redundant in either of two ways:

There is already an axiom implying the existence of at least one set. Given such an axiom together with theaxiom of separation, the existence of the empty set is easily proved.

In the presence of urelements, it is easy to prove that at least one set exists, viz. the set of all urelements. Again,given the axiom of separation, the empty set is easily proved.

5.4.2 Philosophical issues

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity,whose meaning and usefulness are debated by philosophers and logicians.The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something.This issue can be overcome by viewing a set as a bagan empty bag undoubtedly still exists. Darling (2004) explainsthat the empty set is not nothing, but rather the set of all triangles with four sides, the set of all numbers that arebigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king.[4]

The popular syllogism

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sand-wich is better than eternal happiness

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darlingwrites that the contrast can be seen by rewriting the statements Nothing is better than eternal happiness and "[A]ham sandwich is better than nothing in a mathematical tone. According to Darling, the former is equivalent to Theset of all things that are better than eternal happiness is ; " and the latter to The set {ham sandwich} is better thanthe set ; ". It is noted that the rst compares elements of sets, while the second compares the sets themselves.[4]Jonathan Lowe argues that while the empty set:

"...was undoubtedly an important landmark in the history of mathematics, we should not assume thatits utility in calculation is dependent upon its actually denoting some object.

it is also the case that:

All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and(3) is unique amongst sets in having no members. However, there are very many things that 'have nomembers, in the set-theoretical sensenamely, all non-sets. It is perfectly clear why these things haveno members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a setwhich has no members. We cannot conjure such an entity into existence by mere stipulation.[5]

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtainedby plural quantication over individuals, without reifying sets as singular entities having other entities as members.[6]

5.5. SEE ALSO 27

5.5 See also Inhabited set Nothing

5.6 Notes[1] Earliest Uses of Symbols of Set Theory and Logic.

[2] Unicode Standard 5.2

[3] Bruckner, A.N., Bruckner, J.B., and Thomson, B.S., 2008. Elementary Real Analysis, 2nd ed. Prentice Hall. P. 9.

[4] D. J. Darling (2004). The universal book of mathematics. John Wiley and Sons. p. 106. ISBN 0-471-27047-4.

[5] E. J. Lowe (2005). Locke. Routledge. p. 87.

[6] George Boolos, 1984, To be is to be the value of a variable, The Journal of Philosophy 91: 43049. Reprinted inhis 1998 Logic, Logic and Logic (Richard Jerey, and Burgess, J., eds.) Harvard Univ. Press: 5472.

5.7 References Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN3-540-44085-2

Graham, Malcolm (1975), Modern Elementary Mathematics (HARDCOVER) (in English) (2nd ed.), NewYork: Harcourt Brace Jovanovich, ISBN 0155610392

5.8 External links Weisstein, Eric W., Empty Set, MathWorld.

Chapter 6

Family of sets

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family ofsubsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.The term collection is used here because, in some contexts, a family of sets may be allowed to contain repeatedcopies of any given member,[1][2][3] and in other contexts it may form a proper class rather than a set.

6.1 Examples The power set P(S) is a family of sets over S. The k-subsets S(k) of a set S form a family of sets. Let S = {a,b,c,1,2}, an example of a family of sets over S (in the multiset sense) is given by F = {A1, A2, A3,A4} where A1 = {a,b,c}, A2 = {1,2}, A3 = {1,2} and A4 = {a,b,1}.

The class Ord of all ordinal numbers is a large family of sets; that is, it is not itself a set but instead a properclass.

6.2 Special types of set family A Sperner family is a family of sets in which none of the sets contains any of the others. Sperners theorembounds the maximum size of a Sperner family.

A Helly family is a family of sets such that any minimal subfamily with empty intersection has bounded size.Hellys theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families.

6.3 Properties Any family of subsets of S is itself a subset of the power set P(S) if it has no repeated members. Any family of sets without repetitions is a subclass of the proper class V of all sets (the universe). Halls marriage theorem, due to Philip Hall gives necessary and sucient conditions for a nite family ofnon-empty sets (repetitions allowed) to have a system of distinct representatives.

6.4 Related conceptsCertain types of objects from other areas ofmathematics are equivalent to families of sets, in that they can be describedpurely as a collection of sets of objects of some type:

28

6.5. SEE ALSO 29

A hypergraph, also called a set system, is formed by a set of vertices together with another set of hyperedges,each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any familyof sets can be interpreted as a hypergraph that has the union of the sets as its vertices.

An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shapeformed by unions of line segments, triangles, tetrahedra, and higher-dimensional simplices, joined face to face.In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family ofnite sets without repetitions in which the subsets of any set in the family also belong to the family forms anabstract simplicial complex.

An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation, called theincidence relation, specifying which points belong to which lines. An incidence structure can be specied bya family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to eachline, and any family of sets can be interpreted as an incidence structure in this way.

A binary block code consists of a set of codewords, each of which is a string of 0s and 1s, all the same length.When each pair of codewords has large Hamming distance, it can be used as an error-correcting code. A blockcode can also be described as a family of sets, by describing each codeword as the set of positions at which itcontains a 1.

6.5 See also Indexed family Class (set theory) Combinatorial design Russells paradox (or Set of sets that do not contain themselves)

6.6 Notes[1] Brualdi 2010, pg. 322

[2] Roberts & Tesman 2009, pg. 692

[3] Biggs 1985, pg. 89

6.7 References Biggs, Norman L. (1985), Discrete Mathematics, Oxford: Clarendon Press, ISBN 0-19-853252-0 Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Upper Saddle River, NJ: Prentice Hall, ISBN0-13-602040-2

Roberts, Fred S.; Tesman, Barry (2009), Applied Combinatorics (2nd ed.), Boca Raton: CRC Press, ISBN978-1-4200-9982-9

Chapter 7

Intersection (set theory)

Intersection of two sets:A \B

In mathematics, the intersection A B of two sets A and B is the set that contains all elements of A that also belongto B (or equivalently, all elements of B that also belong to A), but no other elements.[1]

For explanation of the symbols used in this article, refer to the table of mathematical symbols.

7.1 Basic denitionThe intersection of A and B is written "A B". Formally:

A \B = fx : x 2 A ^ x 2 Bg

30

7.1. BASIC DEFINITION 31

Intersection of three sets:A \B \ C

that is

x A B if and only if

x A and x B.

For example:

The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}. The number 9 is not in the intersection of the set of prime numbers {2, 3, 5, 7, 11, } and the setof odd numbers {1, 3, 5, 7, 9, 11, }.[2]

More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example,is A B C D = A (B (C D)). Intersection is an associative operation; thus,A (B C) = (A B) C.

32 CHAPTER 7. INTERSECTION (SET THEORY)

Intersections of the Greek, English and Russian alphabet (upper case graphemes)

Inside a universe U one may dene the complement Ac of A to be the set of all elements of U not in A. Now theintersection of A and Bmay be written as the complement of the union of their complements, derived easily from DeMorgans laws:A B = (Ac Bc)c

7.1.1 Intersecting and disjoint sets

We say that A intersects (meets) B at an element x if x belongs to A and B. We say that A intersects (meets) B if Aintersects B at some element. A intersects B if their intersection is inhabited.We say that A and B are disjoint if A does not intersect B. In plain language, they have no elements in common. Aand B are disjoint if their intersection is empty, denoted A \B = ? .For example, the sets {1, 2} and {3, 4} are disjoint, the set of even numbers intersects the set of multiples of 3 at 0,6, 12, 18 and other numbers.

7.2. ARBITRARY INTERSECTIONS 33

Example of an intersection with sets

7.2 Arbitrary intersectionsThe most general notion is the intersection of an arbitrary nonempty collection of sets. IfM is a nonempty set whoseelements are themselves sets, then x is an element of the intersection of M if and only if for every element A of M, xis an element of A. In symbols:

x 2

\M, (8A 2M; x 2 A) :

The notation for this last concept can vary considerably. Set theorists will sometimes write "M", while others willinstead write "AM A". The latter notation can be generalized to "iI Ai", which refers to the intersection of thecollection {Ai : i I}. Here I is a nonempty set, and Ai is a set for every i in I.In the case that the index set I is the set of natural numbers, notation analogous to that of an innite series may be

34 CHAPTER 7. INTERSECTION (SET THEORY)

seen:

1\i=1

Ai:

When formatting is dicult, this can also be written "A1 A2 A3 ..., even though strictly speaking, A1 (A2 (A3 ... makes no sense. (This last example, an intersection of countably many sets, is actually very common; for anexample see the article on -algebras.)Finally, let us note that whenever the symbol "" is placed before other symbols instead of between them, it shouldbe of a larger size ().

7.3 Nullary intersection

Conjunctions of the arguments in parenthesesThe conjunction of no argument is the tautology (compare: empty product); accordingly the intersection of no set is the universe.

Note that in the previous section we excluded the case whereM was the empty set (). The reason is as follows: Theintersection of the collection M is dened as the set (see set-builder notation)

\M = fx : 8A 2M; x 2 Ag:

IfM is empty there are no sets A inM, so the question becomes which x's satisfy the stated condition?" The answerseems to be every possible x. When M is empty the condition given above is an example of a vacuous truth. So theintersection of the empty family should be the universal set (the identity element for the operation of intersection) [3]

7.4. SEE ALSO 35

Unfortunately, according to standard (ZFC) set theory, the universal set does not exist. A partial x for this problemcan be found if we agree to restrict our attention to subsets of a xed set U called the universe. In this case theintersection of a family of subsets of U can be dened as

\M = fx 2 U : 8A 2M; x 2 Ag:

Now if M is empty there is no problem. The intersection is just the entire universe U, which is a well-dened set byassumption and becomes the identity element for this operation.

7.4 See also Complement Intersection graph Logical conjunction Naive set theory Symmetric dierence Union Cardinality Iterated binary operation MinHash

7.5 References[1] Stats: Probability Rules. People.richland.edu. Retrieved 2012-05-08.

[2] How to nd the intersection of sets

[3] Megginson, Robert E. (1998), Chapter 1, An introduction to Banach space theory, Graduate Texts in Mathematics 183,New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3

7.6 Further reading Devlin, K. J. (1993). The Joy of Sets: Fundamentals of Contemporary Set Theory (Second ed.). New York,NY: Springer-Verlag. ISBN 3-540-94094-4.

Munkres, James R. (2000). Set Theory and Logic. Topology (Second ed.). Upper Saddle River: PrenticeHall. ISBN 0-13-181629-2.

Rosen, Kenneth (2007). Basic Structures: Sets, Functions, Sequences, and Sums. Discrete Mathematics andIts Applications (Sixth ed.). Boston: McGraw-Hill. ISBN 978-0-07-322972-0.

7.7 External links Weisstein, Eric W., Intersection, MathWorld.

Chapter 8

Subset

Superset redirects here. For other uses, see Superset (disambiguation).In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is

AB

Euler diagram showingA is a proper subset of B and conversely B is a proper superset of A

36

8.1. DEFINITIONS 37

contained inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of oneset being a subset of another is called inclusion or sometimes containment.The subset relation denes a partial order on sets.The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

8.1 DenitionsIf A and B are sets and every element of A is also an element of B, then:

A is a subset of (or is included in) B, denoted by A B ,or equivalently B is a superset of (or includes) A, denoted by B A:

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A),then

A is also a proper (or strict) subset of B; this is written as A ( B:or equivalently B is a proper superset of A; this is written as B ) A:

For any set S, the inclusion relation is a partial order on the set P(S) of all subsets of S (the power set of S).When quantied, A B is represented as: x{xA xB}.[1]

8.2 and symbolsSome authors use the symbols and to indicate subset and superset respectively; that is, with the same meaningand instead of the symbols, and .[2] So for example, for these authors, it is true of every set A that A A.Other authors prefer to use the symbols and to indicate proper subset and superset, respectively, instead of and.[3] This usage makes and analogous to the inequality symbols and

38 CHAPTER 8. SUBSET

polygonsregular

polygons

The regular polygons form a subset of the polygons

Another example in an Euler diagram:

A is a proper subset of B C is a subset but no proper subset of B

8.4 Other properties of inclusionInclusion is the canonical partial order in the sense that every partially ordered set (X, ) is isomorphic to somecollection of sets ordered by inclusion. The ordinal numbers are a simple exampleif each ordinal n is identiedwith the set [n] of all ordinals less than or equal to n, then a b if and only if [a] [b].For the power set P(S) of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product ofk = |S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumeratingS = {s1, s2, , sk} and associating with each subset T S (which is to say with each element of 2S) the k-tuple from{0,1}k of which the ith coordinate is 1 if and only if si is a member of T.

8.5 See also Containment order

8.6 References[1] Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN

978-0-07-338309-5.

[2] Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1,MR 924157

8.7. EXTERNAL LINKS 39

C B AA B and B C imply A C

[3] Subsets and Proper Subsets (PDF), retrieved 2012-09-07

Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

8.7 External links Weisstein, Eric W., Subset, MathWorld.

Chapter 9

Union (set theory)

Union of two sets:A [B

In set theory, the union (denoted by ) of a collection of sets is the set of all distinct elements in the collection.[1] Itis one of the fundamental operations through which sets can be combined and related to each other.

9.1 Union of two setsThe union of two sets A and B is the set of elements which are in A, in B, or in both A and B. In symbols,

A [B = fx : x 2 A or x 2 Bg

40

9.2. ALGEBRAIC PROPERTIES 41

Union of three sets:A [B [ C

For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6} then A B = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example(involving two innite sets) is:

A = {x is an even integer larger than 1}B = {x is an odd integer larger than 1}A [B = f2; 3; 4; 5; 6; : : : g

Sets cannot have duplicate elements, so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. Multipleoccurrences of identical elements have no eect on the cardinality of a set or its contents.The number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, } and the set of evennumbers {2, 4, 6, 8, 10, }, because 9 is neither prime nor even.

9.2 Algebraic propertiesBinary union is an associative operation; that is,

42 CHAPTER 9. UNION (SET THEORY)

A (B C) = (A B) C.

The operations can be performed in any order, and the parentheses may be omitted without ambiguity (i.e., either ofthe above can be expressed equivalently as A B C). Similarly, union is commutative, so the sets can be written inany order.The empty set is an identity element for the operation of union. That is, A = A, for any set A.These facts follow from analogous facts about logical disjunction.

9.3 Finite unionsOne can take the union of several sets simultaneously. For example, the union of three sets A, B, and C contains allelements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A B C if andonly if x is in at least one of A, B, and C.In mathematics a nite union means any union carried out on a nite number of sets: it doesn't imply that the unionset is a nite set.

9.4 Arbitrary unionsThe most general notion is the union of an arbitrary collection of sets, sometimes called an innitary union. IfM isa set whose elements are themselves sets, then x is an element of the union of M if and only if there is at least oneelement A ofM such that x is an element of A. In symbols:

x 2[

M () 9A 2M; x 2 A:

That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union inaxiomatic set theory.This idea subsumes the preceding sections, in that (for example) A B C is the union of the collection {A,B,C}.Also, if M is the empty collection, then the union of M is the empty set. The analogy between nite unions andlogical disjunction extends to one between arbitrary unions and existential quantication.

9.4.1 NotationsThe notation for the general concept can vary considerably. For a nite union of sets S1; S2; S3; : : : ; Sn one oftenwrites S1 [ S2 [ S3 [ [ Sn . Various common notations for arbitrary unions include

SM , SA2MA , andSi2I Ai , the last of which refers to the union of the collection fAi : i 2 Ig where I is an index set and Ai is a set

for every i 2 I . In the case that the index set I is the set of natural numbers, one uses a notationS1i=1Ai analogousto that of the innite series. When formatting is dicult, this can also be written "A1 A2 A3 ". (This lastexample, a union of countably many sets, is very common in analysis; for an example see the article on -algebras.)Whenever the symbol "" is placed before other symbols instead of between them, it is of a larger size.

9.4.2 Union and intersectionSince sets with unions and intersections form a Boolean algebra, Intersection distributes over union:

A \ (B [ C) = (A \B) [ (A \ C)and union distributes over intersection:

A [ (B \ C) = (A [B) \ (A [ C)

9.5. SEE ALSO 43

Within a given universal set, union can be written in terms of the operations of intersection and complement as

A [B = AC \BCCwhere the superscript C denotes the complement with respect to the universal set.Arbitrary union and intersection also satisfy the law

[i2I

\j2J

Ai;j

\j2J

[i2I

Ai;j

9.5 See also Alternation (formal language theory), the union of sets of strings Cardinality Complement (set theory) Disjoint union Intersection (set theory) Iterated binary operation Naive set theory Symmetric dierence

9.6 Notes[1] Weisstein, Eric W. Union. Wolframs Mathworld. Retrieved 2009-07-14.

9.7 External links Weisstein, Eric W., Union, MathWorld. Hazewinkel, Michiel, ed. (2001), Union of sets, Encyclopedia ofMathematics, Springer, ISBN 978-1-55608-010-4

Innite Union and Intersection at ProvenMath DeMorgans laws formally proven from the axioms of set theory.

44 CHAPTER 9. UNION (SET THEORY)

9.8 Text and image sources, contributors, and licenses9.8.1 Text

Algebra of sets Source: https://en.wikipedia.org/wiki/Algebra_of_sets?oldid=660214875 Contributors: The Anome, AugPi, CharlesMatthews, WhisperToMe, Wile E. Heresiarch, Tobias Bergemann, Macrakis, Doshell, Discospinster, Esse~enwiki, Slipstream, Paul Au-gust, Oleg Alexandrov, Woohookitty, Isnow, Salix alba, Juan Marquez, Mathbot, Splintercellguy, Trovatore, Arthur Rubin, Gilliam,Bluebot, Javalenok, Byelf2007, Jackzhp, Kupirijo, Josephpetty100, Policron, The enemies of god, Alex10023, Jamelan, Tcamps42, An-chor Link Bot, Hans Adler, Addbot, Legobot, TaBOT-zerem, Matt Popat, Bluerasberry, Corruptcopper, Saeidpourbabak, Yahia.barie,Specs112, MegaSloth, DASHBot, Set theorist, ClueBot NG, MerlIwBot, DonMTobin, Helpful Pixie Bot, Daviddwd, Palltrast, Chris-Gualtieri, Freeze S, Jochen Burghardt, YiFeiBot, Eniacpx, Plcarmelbiron and Anonymous: 41

Binary relation Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=669144544 Contributors: AxelBoldt, Bryan Derksen, Zun-dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropu,Dratman, Jorge Stol, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koeyahoo, Trovatore,Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egrin, Rlupsa, JAnD-bot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant,Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin,Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon,ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Ko-ertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, ,Some1Redirects4You and Anonymous: 102

Disjoint sets Source: https://en.wikipedia.org/wiki/Disjoint_sets?oldid=655510316 Contributors: AxelBoldt, Mav, Tarquin, Jeronimo,Shd~enwiki, Arvindn, Toby Bartels, Michael Hardy, Wshun, Revolver, Charles Matthews, Fibonacci, Robbot, Sheskar~enwiki, TobiasBergemann, Giftl