Set Families

Set familiesFrom Wikipedia, the free encyclopedia

Chapter 1

Abstract simplicial complex

In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of asimplicial complex, consisting of a family of non-empty nite sets closed under the operation of taking non-emptysubsets.[1] In the context of matroids and greedoids, abstract simplicial complexes are also called independencesystems.[2]

1.1 DenitionsA family of non-empty nite subsets of a universal set S is an abstract simplicial complex if, for every set X in, and every non-empty subset Y X, Y also belongs to .The nite sets that belong to are called faces of the complex, and a face Y is said to belong to another face X if Y X, so the denition of an abstract simplicial complex can be restated as saying that every face of a face of a complex is itself a face of . The vertex set of is dened as V() = , the union of all faces of . The elements of thevertex set are called the vertices of the complex. So for every vertex v of , the set {v} is a face of the complex.The maximal faces of (i.e., faces that are not subsets of any other faces) are called facets of the complex. Thedimension of a face X in is dened as dim(X) = |X| 1: faces consisting of a single element are zero-dimensional,faces consisting of two elements are one-dimensional, etc. The dimension of the complex dim() is dened as thelargest dimension of any of its faces, or innity if there is no nite bound on the dimension of the faces.The complex is said to be nite if it has nitely many faces, or equivalently if its vertex set is nite. Also, issaid to be pure if it is nite-dimensional (but not necessarily nite) and every facet has the same dimension. In otherwords, is pure if dim() is nite and every face is contained in a facet of dimension dim().One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertexset of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspondto undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that donot have any incident edges.A subcomplex of is a simplicial complex L such that every face of L belongs to ; that is, L and L is asimplicial complex. A subcomplex that consists of all of the subsets of a single face of is often called a simplexof . (However, some authors use the term simplex for a face or, rather ambiguously, for both a face and thesubcomplex associated with a face, by analogy with the non-abstract (geometric) simplicial complex terminology. Toavoid ambiguity, we do not use in this article the term simplex for a face in the context of abstract complexes.)The d-skeleton of is the subcomplex of consisting of all of the faces of that have dimension at most d. Inparticular, the 1-skeleton is called the underlying graph of . The 0-skeleton of can be identied with its vertexset, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the0-skeleton is a family of single-element sets).The link of a face Y in , often denoted /Y or lk(Y), is the subcomplex of dened by

/Y := fX 2 j X \ Y = ?; X [ Y 2 g:Note that the link of the empty set is itself.

2

1.2. GEOMETRIC REALIZATION 3

Given two abstract simplicial complexes, and , a simplicial map is a function f that maps the vertices of to thevertices of and that has the property that for any face X of , the image set f (X) is a face of .

1.2 Geometric realizationWe can associate to an abstract simplicial complex K a topological space |K |, called its geometric realization, whichis a simplicial complex. The construction goes as follows.First, dene |K | as a subset of [0, 1]S consisting of functions t : S [0, 1] satisfying the two conditions:

Xs2S

ts = 1

fs 2 S : ts > 0g 2 Now think of [0, 1]S as the direct limit of [0, 1]A where A ranges over nite subsets of S, and give [0, 1]S the inducedtopology. Now give |K | the subspace topology.Alternatively, let K denote the category whose objects are the faces of K and whose morphisms are inclusions. Nextchoose a total order on the vertex set of K and dene a functor F from K to the category of topological spaces asfollows. For any face X K of dimension n, let F(X) = n be the standard n-simplex. The order on the vertex setthen species a unique bijection between the elements of X and vertices of n, ordered in the usual way e0 < e1 < ...< en. If Y X is a face of dimension m < n, then this bijection species a unique m-dimensional face of n. DeneF(Y) F(X) to be the unique ane linear embedding of m as that distinguished face of n, such that the map onvertices is order preserving.We can then dene the geometric realization |K | as the colimit of the functor F. More specically |K | is the quotientspace of the disjoint union

aX2K

F (X)

by the equivalence relation which identies a point y F(Y) with its image under the map F(Y) F(X), for everyinclusion Y X.If K is nite, then we can describe |K | more simply. Choose an embedding of the vertex set of K as an anelyindependent subset of some Euclidean space RN of suciently high dimension N. Then any face X K can beidentied with the geometric simplex in RN spanned by the corresponding embedded vertices. Take |K | to be theunion of all such simplices.If K is the standard combinatorial n-simplex, then |K | can be naturally identied with n.

1.3 Examples As an example, let V be a nite subset of S of cardinality n + 1 and let K be the power set of V. Then K is calleda combinatorial n-simplex with vertex set V. If V = S = {0, 1, ..., n}, K is called the standard combinatorialn-simplex.

The clique complex of an undirected graph has a simplex for each clique (complete subgraph) of the givengraph. Clique complexes form the prototypical example of ag complexes, complexes with the property thatevery set of elements that pairwise belong to simplexes of the complex is itself a simplex.

In the theory of partially ordered sets (posets), the order complex of a poset is the set of all nite chains.Its homology groups and other topological invariants contain important information about the poset.

The VietorisRips complex is dened from any metric spaceM and distance by forming a simplex for everynite subset of M with diameter at most . It has applications in homology theory, hyperbolic groups, imageprocessing, and mobile ad hoc networking. It is another example of a ag complex.

4 CHAPTER 1. ABSTRACT SIMPLICIAL COMPLEX

1.4 EnumerationThe number of abstract simplicial complexes on up to n elements is one less than the nth Dedekind number. Thesenumbers grow very rapidly, and are known only for n 8; they are (starting with n = 0):

1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787 (sequence A014466 inOEIS). This corresponds to the number of nonempty antichains of subsets of an n set.

The number of abstract simplicial complexes on exactly n labeled elements is given by the sequence 1, 2, 9, 114,6894, 7785062, 2414627396434, 56130437209370320359966 (sequence A006126 in OEIS), starting at n = 1.This corresponds to the number of antichain covers of a labeled n-set; there is a clear bijection between antichaincovers of an n-set and simplicial complexes on n elements described in terms of their maximal faces.The number of abstract simplicial complexes on exactly n unlabeled elements is given by the sequence 1, 2, 5, 20,180, 16143 (sequence A006602 in OEIS) , starting at n = 1.

1.5 See also KruskalKatona theorem

1.6 References[1] Lee, JM, Introduction to Topological Manifolds, Springer 2011, ISBN 1-4419-7939-5, p153

[2] Korte, Bernhard; Lovsz, Lszl; Schrader, Rainer (1991). Greedoids. Springer-Verlag. p. 9. ISBN 3-540-18190-3.

1.6. REFERENCES 5

A geometrical representation of an abstract simplicial complex that is not a valid simplicial complex.

Chapter 2

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.

2.1 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.

2.2 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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2.3. THE PRINCIPLE OF DUALITY 7

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.

2.3 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.

2.4 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

8 CHAPTER 2. ALGEBRA OF SETS

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)

2.5 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

2.6 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:

2.7. THE ALGEBRA OF RELATIVE COMPLEMENTS 9

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.

2.7 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 = ?

10 CHAPTER 2. ALGEBRA OF SETS

2.8 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)

2.9 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

2.10 External links Operations on Sets at ProvenMath

Chapter 3

Almost disjoint sets

In mathematics, two sets are almost disjoint [1][2] if their intersection is small in some sense; dierent denitions ofsmall will result in dierent denitions of almost disjoint.

3.1 DenitionThemost common choice is to take small to mean nite. In this case, two sets are almost disjoint if their intersectionis nite, i.e. if

jA \Bj

12 CHAPTER 3. ALMOST DISJOINT SETS

3.2 Other meaningsSometimes almost disjoint is used in some other sense, or in the sense of measure theory or topological category.Here are some alternative denitions of almost disjoint that are sometimes used (similar denitions apply to innitecollections):

Let be any cardinal number. Then two sets A and B are almost disjoint if the cardinality of their intersectionis less than , i.e. if

jA \Bj < :

The case of = 1 is simply the denition of disjoint sets; the case of

= @0

is simply the denition of almost disjoint given above, where the intersection of A and B is nite.

Let m be a complete measure on a measure space X. Then two subsets A and B of X are almost disjoint if theirintersection is a null-set, i.e. if

m(A \B) = 0:

Let X be a topological space. Then two subsets A and B of X are almost disjoint if their intersection is meagrein X.

3.3 References[1] Kunen, K. (1980), Set Theory; an introduction to independence proofs, North Holland, p. 47

[2] Jech, R. (2006) Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118

Chapter 4

Antimatroid

{a,b}

{a,b,c}

{a,c} {b,c}

{a} {c}

abcaba

acacbccacabcbcba

{a} {c}

{a,b} {b,c}

{a,b,c,d}

abcd

acbd

cabd

cbad

{a,b,c,d}

Three views of an antimatroid: an inclusion ordering on its family of feasible sets, a formal language, and the corresponding pathposet.

In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by includingelements one at a time, and in which an element, once available for inclusion, remains available until it is included.Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible statesof such a process, or as a formal language modeling the dierent sequences in which elements may be included.Dilworth (1940) was the rst to study antimatroids, using yet another axiomatization based on lattice theory, andthey have been frequently rediscovered in other contexts;[1] see Korte et al. (1991) for a comprehensive survey ofantimatroid theory with many additional references.The axioms dening antimatroids as set systems are very similar to those ofmatroids, but whereasmatroids are denedby an exchange axiom (e.g., the basis exchange, or independent set exchange axioms), antimatroids are dened insteadby an anti-exchange axiom, from which their name derives. Antimatroids can be viewed as a special case of greedoidsand of semimodular lattices, and as a generalization of partial orders and of distributive lattices. Antimatroids areequivalent, by complementation, to convex geometries, a combinatorial abstraction of convex sets in geometry.

13

14 CHAPTER 4. ANTIMATROID

Antimatroids have been applied to model precedence constraints in scheduling problems, potential event sequencesin simulations, task planning in articial intelligence, and the states of knowledge of human learners.

4.1 DenitionsAn antimatroid can be dened as a nite family F of sets, called feasible sets, with the following two properties:

The union of any two feasible sets is also feasible. That is, F is closed under unions.

If S is a nonempty feasible set, then there exists some x in S such that S \ {x} (the set formed by removing xfrom S) is also feasible. That is, F is an accessible set system.

Antimatroids also have an equivalent denition as a formal language, that is, as a set of strings dened from a nitealphabet of symbols. A language L dening an antimatroid must satisfy the following properties:

Every symbol of the alphabet occurs in at least one word of L.

Each word of L contains at most one copy of any symbol.

Every prex of a string in L is also in L.

If s and t are strings in L, and s contains at least one symbol that is not in t, then there is a symbol x in s suchthat tx is another string in L.

If L is an antimatroid dened as a formal language, then the sets of symbols in strings of L form an accessible union-closed set system. In the other direction, if F is an accessible union-closed set system, and L is the language of stringss with the property that the set of symbols in each prex of s is feasible, then L denes an antimatroid. Thus, thesetwo denitions lead to mathematically equivalent classes of objects.[2]

4.2 Examples A chain antimatroid has as its formal language the prexes of a single word, and as its feasible sets the setsof symbols in these prexes. For instance the chain antimatroid dened by the word abcd has as its formallanguage the strings {, a, ab, abc, abcd"} and as its feasible sets the sets , {a}, {a,b}, {a,b,c}, and{a,b,c,d}.[3]

A poset antimatroid has as its feasible sets the lower sets of a nite partially ordered set. By Birkhos rep-resentation theorem for distributive lattices, the feasible sets in a poset antimatroid (ordered by set inclusion)form a distributive lattice, and any distributive lattice can be formed in this way. Thus, antimatroids can beseen as generalizations of distributive lattices. A chain antimatroid is the special case of a poset antimatroidfor a total order.[3]

A shelling sequence of a nite set U of points in the Euclidean plane or a higher-dimensional Euclidean spaceis an ordering on the points such that, for each point p, there is a line (in the Euclidean plane, or a hyperplanein a Euclidean space) that separates p from all later points in the sequence. Equivalently, p must be a vertexof the convex hull of it and all later points. The partial shelling sequences of a point set form an antimatroid,called a shelling antimatroid. The feasible sets of the shelling antimatroid are the intersections of U with thecomplement of a convex set.[3]

A perfect elimination ordering of a chordal graph is an ordering of its vertices such that, for each vertex v,the neighbors of v that occur later than v in the ordering form a clique. The prexes of perfect eliminationorderings of a chordal graph form an antimatroid.[3]

4.3. PATHS AND BASIC WORDS 15

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

A shelling sequence of a planar point set. The line segments show edges of the convex hulls after some of the points have beenremoved.

4.3 Paths and basic words

In the set theoretic axiomatization of an antimatroid there are certain special sets called paths that determine thewhole antimatroid, in the sense that the sets of the antimatroid are exactly the unions of paths. If S is any feasible setof the antimatroid, an element x that can be removed from S to form another feasible set is called an endpoint of S,and a feasible set that has only one endpoint is called a path of the antimatroid. The family of paths can be partiallyordered by set inclusion, forming the path poset of the antimatroid.For every feasible set S in the antimatroid, and every element x of S, one may nd a path subset of S for which xis an endpoint: to do so, remove one at a time elements other than x until no such removal leaves a feasible subset.Therefore, each feasible set in an antimatroid is the union of its path subsets. If S is not a path, each subset in thisunion is a proper subset of S. But, if S is itself a path with endpoint x, each proper subset of S that belongs to theantimatroid excludes x. Therefore, the paths of an antimatroid are exactly the sets that do not equal the unions oftheir proper subsets in the antimatroid. Equivalently, a given family of sets P forms the set of paths of an antimatroidif and only if, for each S in P, the union of subsets of S in P has one fewer element than S itself. If so, F itself is thefamily of unions of subsets of P.In the formal language formalization of an antimatroid wemay also identify a subset of words that determine the wholelanguage, the basic words. The longest strings in L are called basic words; each basic word forms a permutation ofthe whole alphabet. For instance, the basic words of a poset antimatroid are the linear extensions of the given partialorder. If B is the set of basic words, L can be dened from B as the set of prexes of words in B. It is often convenientto dene antimatroids from basic words in this way, but it is not straightforward to write an axiomatic denition of

16 CHAPTER 4. ANTIMATROID

antimatroids in terms of their basic words.

4.4 Convex geometriesSee also: Convex set, Convex geometry and Closure operator

If F is the set system dening an antimatroid, with U equal to the union of the sets in F, then the family of sets

G = fU n S j S 2 Fg

complementary to the sets in F is sometimes called a convex geometry and the sets in G are called convex sets. Forinstance, in a shelling antimatroid, the convex sets are intersections of U with convex subsets of the Euclidean spaceinto which U is embedded.Complementarily to the properties of set systems that dene antimatroids, the set system dening a convex geometryshould be closed under intersections, and for any set S in G that is not equal to U there must be an element x not in Sthat can be added to S to form another set in G.A convex geometry can also be dened in terms of a closure operator that maps any subset of U to its minimalclosed superset. To be a closure operator, should have the following properties:

() = : the closure of the empty set is empty. Any set S is a subset of (S). If S is a subset of T, then (S) must be a subset of (T). For any set S, (S) = ((S)).

The family of closed sets resulting from a closure operation of this type is necessarily closed under intersections. Theclosure operators that dene convex geometries also satisfy an additional anti-exchange axiom:

If neither y nor z belong to (S), but z belongs to (S {y}), then y does not belong to (S {z}).

A closure operation satisfying this axiom is called an anti-exchange closure. If S is a closed set in an anti-exchangeclosure, then the anti-exchange axiom determines a partial order on the elements not belonging to S, where x y inthe partial order when x belongs to (S {y}). If x is a minimal element of this partial order, then S {x} is closed.That is, the family of closed sets of an anti-exchange closure has the property that for any set other than the universalset there is an element x that can be added to it to produce another closed set. This property is complementary tothe accessibility property of antimatroids, and the fact that intersections of closed sets are closed is complementaryto the property that unions of feasible sets in an antimatroid are feasible. Therefore, the complements of the closedsets of any anti-exchange closure form an antimatroid.[4]

4.5 Join-distributive latticesAny two sets in an antimatroid have a unique least upper bound (their union) and a unique greatest lower bound(the union of the sets in the antimatroid that are contained in both of them). Therefore, the sets of an antimatroid,partially ordered by set inclusion, form a lattice. Various important features of an antimatroid can be interpreted inlattice-theoretic terms; for instance the paths of an antimatroid are the join-irreducible elements of the correspondinglattice, and the basic words of the antimatroid correspond to maximal chains in the lattice. The lattices that arise fromantimatroids in this way generalize the nite distributive lattices, and can be characterized in several dierent ways.

The description originally considered by Dilworth (1940) concerns meet-irreducible elements of the lattice.For each element x of an antimatroid, there exists a unique maximal feasible set Sx that does not contain x (Sxis the union of all feasible sets not containing x). Sx is meet-irreducible, meaning that it is not the meet of any

4.6. SUPERSOLVABLE ANTIMATROIDS 17

two larger lattice elements: any larger feasible set, and any intersection of larger feasible sets, contains x and sodoes not equal Sx. Any element of any lattice can be decomposed as a meet of meet-irreducible sets, often inmultiple ways, but in the lattice corresponding to an antimatroid each element T has a unique minimal familyof meet-irreducible sets Sx whose meet is T ; this family consists of the sets Sx such that T {x} belongs to theantimatroid. That is, the lattice has unique meet-irreducible decompositions.

A second characterization concerns the intervals in the lattice, the sublattices dened by a pair of lattice elementsx y and consisting of all lattice elements z with x z y. An interval is atomistic if every element in it is thejoin of atoms (the minimal elements above the bottom element x), and it is Boolean if it is isomorphic to thelattice of all subsets of a nite set. For an antimatroid, every interval that is atomistic is also boolean.

Thirdly, the lattices arising from antimatroids are semimodular lattices, lattices that satisfy the upper semimod-ular law that for any two elements x and y, if y covers x y then x y covers x. Translating this conditioninto the sets of an antimatroid, if a set Y has only one element not belonging to X then that one element maybe added to X to form another set in the antimatroid. Additionally, the lattice of an antimatroid has the meet-semidistributive property: for all lattice elements x, y, and z, if x y and x z are both equal then they alsoequal x (y z). A semimodular and meet-semidistributive lattice is called a join-distributive lattice.

These three characterizations are equivalent: any lattice with unique meet-irreducible decompositions has booleanatomistic intervals and is join-distributive, any lattice with boolean atomistic intervals has unique meet-irreducibledecompositions and is join-distributive, and any join-distributive lattice has unique meet-irreducible decompositionsand boolean atomistic intervals.[5] Thus, wemay refer to a lattice with any of these three properties as join-distributive.Any antimatroid gives rise to a nite join-distributive lattice, and any nite join-distributive lattice comes from anantimatroid in this way.[6] Another equivalent characterization of nite join-distributive lattices is that they are graded(any two maximal chains have the same length), and the length of a maximal chain equals the number of meet-irreducible elements of the lattice.[7] The antimatroid representing a nite join-distributive lattice can be recoveredfrom the lattice: the elements of the antimatroid can be taken to be the meet-irreducible elements of the lattice, andthe feasible set corresponding to any element x of the lattice consists of the set of meet-irreducible elements y suchthat y is not greater than or equal to x in the lattice.This representation of any nite join-distributive lattice as an accessible family of sets closed under unions (that is, asan antimatroid) may be viewed as an analogue of Birkhos representation theorem under which any nite distributivelattice has a representation as a family of sets closed under unions and intersections.

4.6 Supersolvable antimatroidsMotivated by a problem of dening partial orders on the elements of a Coxeter group, Armstrong (2007) studied an-timatroids which are also supersolvable lattices. A supersolvable antimatroid is dened by a totally ordered collectionof elements, and a family of sets of these elements. The family must include the empty set. Additionally, it musthave the property that if two sets A and B belong to the family, the set-theoretic dierence B \ A is nonempty, and xis the smallest element of B \ A, then A {x} also belongs to the family. As Armstrong observes, any family of setsof this type forms an antimatroid. Armstrong also provides a lattice-theoretic characterization of the antimatroidsthat this construction can form.

4.7 Join operation and convex dimensionIf A and B are two antimatroids, both described as a family of sets, and if the maximal sets in A and B are equal, wecan form another antimatroid, the join of A and B, as follows:

A _B = fS [ T j S 2 A ^ T 2 Bg:

This is a dierent operation than the join considered in the lattice-theoretic characterizations of antimatroids: itcombines two antimatroids to form another antimatroid, rather than combining two sets in an antimatroid to formanother set. The family of all antimatroids that have a given maximal set forms a semilattice with this join operation.

18 CHAPTER 4. ANTIMATROID

Joins are closely related to a closure operation that maps formal languages to antimatroids, where the closure of alanguage L is the intersection of all antimatroids containing L as a sublanguage. This closure has as its feasible setsthe unions of prexes of strings in L. In terms of this closure operation, the join is the closure of the union of thelanguages of A and B.Every antimatroid can be represented as a join of a family of chain antimatroids, or equivalently as the closure ofa set of basic words; the convex dimension of an antimatroid A is the minimum number of chain antimatroids (orequivalently the minimum number of basic words) in such a representation. If F is a family of chain antimatroidswhose basic words all belong to A, then F generates A if and only if the feasible sets of F include all paths of A. Thepaths of A belonging to a single chain antimatroid must form a chain in the path poset of A, so the convex dimensionof an antimatroid equals the minimum number of chains needed to cover the path poset, which by Dilworths theoremequals the width of the path poset.[8]

If one has a representation of an antimatroid as the closure of a set of d basic words, then this representation canbe used to map the feasible sets of the antimatroid into d-dimensional Euclidean space: assign one coordinate perbasic word w, and make the coordinate value of a feasible set S be the length of the longest prex of w that is asubset of S. With this embedding, S is a subset of T if and only if the coordinates for S are all less than or equal tothe corresponding coordinates of T. Therefore, the order dimension of the inclusion ordering of the feasible sets isat most equal to the convex dimension of the antimatroid.[9] However, in general these two dimensions may be verydierent: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.

4.8 EnumerationThe number of possible antimatroids on a set of elements grows rapidly with the number of elements in the set. Forsets of one, two, three, etc. elements, the number of distinct antimatroids is

1, 3, 22, 485, 59386, 133059751, ... (sequence A119770 in OEIS).

4.9 ApplicationsBoth the precedence and release time constraints in the standard notation for theoretic scheduling problems maybe modeled by antimatroids. Boyd & Faigle (1990) use antimatroids to generalize a greedy algorithm of EugeneLawler for optimally solving single-processor scheduling problems with precedence constraints in which the goal isto minimize the maximum penalty incurred by the late scheduling of a task.Glasserman & Yao (1994) use antimatroids to model the ordering of events in discrete event simulation systems.Parmar (2003) uses antimatroids to model progress towards a goal in articial intelligence planning problems.In mathematical psychology, antimatroids have been used to describe feasible states of knowledge of a human learner.Each element of the antimatroid represents a concept that is to be understood by the learner, or a class of problems thathe or she might be able to solve correctly, and the sets of elements that form the antimatroid represent possible sets ofconcepts that could be understood by a single person. The axioms dening an antimatroid may be phrased informallyas stating that learning one concept can never prevent the learner from learning another concept, and that any feasiblestate of knowledge can be reached by learning a single concept at a time. The task of a knowledge assessment systemis to infer the set of concepts known by a given learner by analyzing his or her responses to a small and well-chosenset of problems. In this context antimatroids have also been called learning spaces and well-graded knowledgespaces.[10]

4.10 Notes[1] Two early references are Edelman (1980) and Jamison (1980); Jamison was the rst to use the term antimatroid.

Monjardet (1985) surveys the history of rediscovery of antimatroids.

[2] Korte et al., Theorem 1.4.

4.11. REFERENCES 19

[3] Gordon (1997) describes several results related to antimatroids of this type, but these antimatroids were mentioned earliere.g. by Korte et al. Chandran et al. (2003) use the connection to antimatroids as part of an algorithm for eciently listingall perfect elimination orderings of a given chordal graph.

[4] Korte et al., Theorem 1.1.[5] Adaricheva, Gorbunov & Tumanov (2003), Theorems 1.7 and 1.9; Armstrong (2007), Theorem 2.7.[6] Edelman (1980), Theorem 3.3; Armstrong (2007), Theorem 2.8.[7] Monjardet (1985) credits a dual form of this characterization to several papers from the 1960s by S. P. Avann.[8] Edelman & Saks (1988); Korte et al., Theorem 6.9.[9] Korte et al., Corollary 6.10.[10] Doignon & Falmagne (1999).

4.11 References Adaricheva, K. V.; Gorbunov, V. A.; Tumanov, V. I. (2003), Join-semidistributive lattices and convex ge-ometries, Advances in Mathematics 173 (1): 149, doi:10.1016/S0001-8708(02)00011-7.

Armstrong, Drew (2007), The sorting order on a Coxeter group, arXiv:0712.1047. Birkho, Garrett; Bennett, M.K. (1985), The convexity lattice of a poset,Order 2 (3): 223242, doi:10.1007/BF00333128. Bjrner, Anders; Ziegler, Gnter M. (1992), Introduction to greedoids, in White, Neil,Matroid Applications,Encyclopedia of Mathematics and its Applications 40, Cambridge: Cambridge University Press, pp. 284357,doi:10.1017/CBO9780511662041.009, ISBN 0-521-38165-7, MR 1165537

Boyd, E. Andrew; Faigle, Ulrich (1990), An algorithmic characterization of antimatroids, Discrete AppliedMathematics 28 (3): 197205, doi:10.1016/0166-218X(90)90002-T.

Chandran, L. S.; Ibarra, L.; Ruskey, F.; Sawada, J. (2003), Generating and characterizing the perfect elimina-tion orderings of a chordal graph (PDF), Theoretical Computer Science 307 (2): 303317, doi:10.1016/S0304-3975(03)00221-4.

Dilworth, Robert P. (1940), Lattices with unique irreducible decompositions, Annals of Mathematics 41 (4):771777, doi:10.2307/1968857, JSTOR 1968857.

Doignon, Jean-Paul; Falmagne, Jean-Claude (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2.

Edelman, Paul H. (1980), Meet-distributive lattices and the anti-exchange closure, Algebra Universalis 10(1): 290299, doi:10.1007/BF02482912.

Edelman, Paul H.; Saks, Michael E. (1988), Combinatorial representation and convex dimension of convexgeometries, Order 5 (1): 2332, doi:10.1007/BF00143895.

Glasserman, Paul; Yao, David D. (1994), Monotone Structure in Discrete Event Systems, Wiley Series in Prob-ability and Statistics, Wiley Interscience, ISBN 978-0-471-58041-6.

Gordon, Gary (1997), A invariant for greedoids and antimatroids, Electronic Journal of Combinatorics 4(1): Research Paper 13, MR 1445628.

Jamison, Robert (1980), Copoints in antimatroids, Proceedings of the Eleventh Southeastern Conference onCombinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1980), Vol. II, Con-gressus Numerantium 29, pp. 535544, MR 608454.

Korte, Bernhard; Lovsz, Lszl; Schrader, Rainer (1991), Greedoids, Springer-Verlag, pp. 1943, ISBN3-540-18190-3.

Monjardet, Bernard (1985), A use for frequently rediscovering a concept,Order 1 (4): 415417, doi:10.1007/BF00582748. Parmar, Aarati (2003), Some Mathematical Structures Underlying Ecient Planning, AAAI Spring Sympo-sium on Logical Formalization of Commonsense Reasoning (PDF).

Chapter 5

Bijection

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A bijective function, f: X Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D.

In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elementsof two sets, where every element of one set is paired with exactly one element of the other set, and every elementof the other set is paired with exactly one element of the rst set. There are no unpaired elements. In mathematical

20

5.1. DEFINITION 21

terms, a bijective function f: X Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.A bijection from the setX to the set Y has an inverse function from Y toX. IfX and Y are nite sets, then the existenceof a bijection means they have the same number of elements. For innite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of innite sets.A bijective function from a set to itself is also called a permutation.Bijective functions are essential tomany areas ofmathematics including the denitions of isomorphism, homeomorphism,dieomorphism, permutation group, and projective map.

5.1 DenitionFor more details on notation, see Function (mathematics) Notation.

For a pairing between X and Y (where Y need not be dierent from X) to be a bijection, four properties must hold:

1. each element of X must be paired with at least one element of Y,

2. no element of X may be paired with more than one element of Y,

3. each element of Y must be paired with at least one element of X, and

4. no element of Y may be paired with more than one element of X.

Satisfying properties (1) and (2) means that a bijection is a function with domain X. It is more common to seeproperties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y.Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions).Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injectivefunctions).[1] With this terminology, a bijection is a function which is both a surjection and an injection, or usingother words, a bijection is a function which is both one-to-one and onto.

5.2 Examples

5.2.1 Batting line-up of a baseball or cricket teamConsider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where everyplayer holds a specic spot in a line-up). The set X will be the players on the team (of size nine in the case of baseball)and the set Y will be the positions in the batting order (1st, 2nd, 3rd, etc.) The pairing is given by which playeris in what position in this order. Property (1) is satised since each player is somewhere in the list. Property (2) issatised since no player bats in two (or more) positions in the order. Property (3) says that for each position in theorder, there is some player batting in that position and property (4) states that two or more players are never battingin the same position in the list.

5.2.2 Seats and students of a classroomIn a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks themall to be seated. After a quick look around the room, the instructor declares that there is a bijection between the setof students and the set of seats, where each student is paired with the seat they are sitting in. What the instructorobserved in order to reach this conclusion was that:

1. Every student was in a seat (there was no one standing),

2. No student was in more than one seat,

3. Every seat had someone sitting there (there were no empty seats), and

22 CHAPTER 5. BIJECTION

4. No seat had more than one student in it.

The instructor was able to conclude that there were just as many seats as there were students, without having to counteither set.

5.3 More mathematical examples and some non-examples For any set X, the identity function 1X: X X, 1X(x) = x, is bijective. The function f: R R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y 1)/2 such that f(x)= y. In more generality, any linear function over the reals, f: R R, f(x) = ax + b (where a is non-zero) is abijection. Each real number y is obtained from (paired with) the real number x = (y - b)/a.

The function f: R (-/2, /2), given by f(x) = arctan(x) is bijective since each real number x is pairedwith exactly one angle y in the interval (-/2, /2) so that tan(y) = x (that is, y = arctan(x)). If the codomain(-/2, /2) was made larger to include an integer multiple of /2 then this function would no longer be onto(surjective) since there is no real number which could be paired with the multiple of /2 by this arctan function.

The exponential function, g: R R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) =1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbersR+ (0;+1) , then g becomes bijective; its inverse (see below) is the natural logarithm function ln.

The function h: R R+, h(x) = x2 is not bijective: for instance, h(1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to R+0 [0;+1) , then h becomes bijective; its inverseis the positive square root function.

5.4 InversesA bijection f with domain X (functionally indicated by f: X Y) also denes a relation starting in Y and goingto X (by turning the arrows around). The process of turning the arrows around for an arbitrary function does not,in general, yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function withdomain Y. Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is,the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible. Afunction is invertible if and only if it is a bijection.Stated in concise mathematical notation, a function f: X Y is bijective if and only if it satises the condition

for every y in Y there is a unique x in X with y = f(x).

Continuing with the baseball batting line-up example, the function that is being dened takes as input the name ofone of the players and outputs the position of that player in the batting order. Since this function is a bijection, it hasan inverse function which takes as input a position in the batting order and outputs the player who will be batting inthat position.

5.5 CompositionThe composition g f of two bijections f: X Y and g: Y Z is a bijection. The inverse of g f is (g f)1 =(f1) (g1) .Conversely, if the composition g f of two functions is bijective, we can only say that f is injective and g is surjective.

5.6 Bijections and cardinalityIf X and Y are nite sets, then there exists a bijection between the two sets X and Y if and only if X and Y havethe same number of elements. Indeed, in axiomatic set theory, this is taken as the denition of same number of

5.7. PROPERTIES 23

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A bijection composed of an injection (left) and a surjection (right).

elements (equinumerosity), and generalising this denition to innite sets leads to the concept of cardinal number,a way to distinguish the various sizes of innite sets.

5.7 Properties A function f: R R is bijective if and only if its graph meets every horizontal and vertical line exactly once. If X is a set, then the bijective functions from X to itself, together with the operation of functional composition(), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).

Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of thecodomain with cardinality |B|, one has the following equalities:

|f(A)| = |A| and |f1(B)| = |B|. If X and Y are nite sets with the same cardinality, and f: X Y, then the following are equivalent:

1. f is a bijection.2. f is a surjection.3. f is an injection.

For a nite set S, there is a bijection between the set of possible total orderings of the elements and the set ofbijections from S to S. That is to say, the number of permutations of elements of S is the same as the numberof total orderings of that setnamely, n!.

5.8 Bijections and category theoryBijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are notalways the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphismsmust be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphismswhich are bijective homomorphisms.

24 CHAPTER 5. BIJECTION

5.9 Generalization to partial functionsThe notion of one-one correspondence generalizes to partial functions, where they are called partial bijections,although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partialfunction is already undened for a portion of its domain; thus there is no compelling reason to constrain its inverseto be a total function, i.e. dened everywhere on its domain. The set of all partial bijections on a given base set iscalled the symmetric inverse semigroup.[2]

Another way of dening the same notion is to say that a partial bijection from A to B is any relation R (which turnsout to be a partial function) with the property that R is the graph of a bijection f:AB, where A is a subset of Aand B is a subset of B.[3]

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] Anexample is theMbius transformation simply dened on the complex plane, rather than its completion to the extendedcomplex plane.[5]

5.10 Contrast withThis list is incomplete; you can help by expanding it.

Multivalued function

5.11 See also Injective function Surjective function Bijection, injection and surjection Symmetric group Bijective numeration Bijective proof Cardinality Category theory AxGrothendieck theorem

5.12 Notes[1] There are names associated to properties (1) and (2) as well. A relation which satises property (1) is called a total relation

and a relation satisfying (2) is a single valued relation.

[2] Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[3] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge UniversityPress. p. 289. ISBN 978-0-521-44179-7.

[4] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.

[5] John Meakin (2007). Groups and semigroups: connections and contrasts. In C.M. Campbell, M.R. Quick, E.F. Robert-son, G.C. Smith. Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4.preprint citing Lawson,M.V. (1998). TheMbius InverseMonoid. Journal of Algebra 200 (2): 428. doi:10.1006/jabr.1997.7242.

5.13. REFERENCES 25

5.13 ReferencesThis topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory.Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic maybe found in any of these:

Wolf (1998). Proof, Logic and Conjecture: A Mathematicians Toolbox. Freeman. Sundstrom (2003). Mathematical Reasoning: Writing and Proof. Prentice-Hall. Smith; Eggen; St.Andre (2006). A Transition to Advanced Mathematics (6th Ed.). Thomson (Brooks/Cole). Schumacher (1996). Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley. O'Leary (2003). The Structure of Proof: With Logic and Set Theory. Prentice-Hall. Morash. Bridge to Abstract Mathematics. Random House. Maddox (2002). Mathematical Thinking and Writing. Harcourt/ Academic Press. Lay (2001). Analysis with an introduction to proof. Prentice Hall. Gilbert; Vanstone (2005). An Introduction to Mathematical Thinking. Pearson Prentice-Hall. Fletcher; Patty. Foundations of Higher Mathematics. PWS-Kent. Iglewicz; Stoyle. An Introduction to Mathematical Reasoning. MacMillan. Devlin, Keith (2004). Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/CRC Press.

D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall. Cupillari. The Nuts and Bolts of Proofs. Wadsworth. Bond. Introduction to Abstract Mathematics. Brooks/Cole. Barnier; Feldman (2000). Introduction to Advanced Mathematics. Prentice Hall. Ash. A Primer of Abstract Mathematics. MAA.

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

Weisstein, Eric W., Bijection, MathWorld. Earliest Uses of Some of theWords ofMathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.

Chapter 6

Bijection, injection and surjection

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in whicharguments (input expressions from the domain) and images (output expressions from the codomain) are related ormapped to each other.A function maps elements from its domain to elements in its codomain. Given a function f : A! B

The function is injective (one-to-one) if every element of the codomain is mapped to by at most one elementof the domain. An injective function is an injection. Notationally:

8x; y 2 A; f(x) = f(y)) x = y:Or, equivalently (using logical transposition),8x; y 2 A; x 6= y ) f(x) 6= f(y):

The function is surjective (onto) if every element of the codomain is mapped to by at least one element of thedomain. (That is, the image and the codomain of the function are equal.) A surjective function is a surjection.Notationally:

8y 2 B; 9x 2 A that such y = f(x):

The function is bijective (one-to-one and onto or one-to-one correspondence) if every element of thecodomain is mapped to by exactly one element of the domain. (That is, the function is both injective andsurjective.) A bijective function is a bijection.

An injective function need not be surjective (not all elements of the codomain may be associated with arguments),and a surjective function need not be injective (some images may be associated with more than one argument). Thefour possible combinations of injective and surjective features are illustrated in the diagrams to the right.

6.1 InjectionMain article: Injective functionFor more details on notation, see Function (mathematics) Notation.A function is injective (one-to-one) if every possible element of the codomain is mapped to by at most one argument.Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is aninjection. The formal denition is the following.

The function f : A! B is injective i for all a; b 2 A , we have f(a) = f(b)) a = b:

A function f : A B is injective if and only if A is empty or f is left-invertible; that is, there is a function g :f(A) A such that g o f = identity function on A. Here f(A) is the image of f.

26

6.2. SURJECTION 27

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Injective composition: the second function need not be injective.

Since every function is surjective when its codomain is restricted to its image, every injection induces a bijectiononto its image. More precisely, every injection f : A B can be factored as a bijection followed by an inclusionas follows. Let fR : A f(A) be f with codomain restricted to its image, and let i : f(A) B be the inclusionmap from f(A) into B. Then f = i o fR. A dual factorisation is given for surjections below.

The composition of two injections is again an injection, but if g o f is injective, then it can only be concludedthat f is injective. See the gure at right.

Every embedding is injective.

6.2 SurjectionMain article: Surjective functionA function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, everyelement in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to itscodomain. A surjective function is a surjection. The formal denition is the following.

The function f : A! B is surjective i for all b 2 B , there is a 2 A such that f(a) = b:

A function f : A B is surjective if and only if it is right-invertible, that is, if and only if there is a function g:B A such that f o g = identity function on B. (This statement is equivalent to the axiom of choice.)

By collapsing all arguments mapping to a given xed image, every surjection induces a bijection dened on aquotient of its domain. More precisely, every surjection f : A B can be factored as a non-bijection followedby a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) : A A/~ be theprojection map which sends each x in A to its equivalence class [x]~, and let fP : A/~ B be the well-denedfunction given by fP([x]~) = f(x). Then f = fP o P(~). A dual factorisation is given for injections above.

The composition of two surjections is again a surjection, but if g o f is surjective, then it can only be concludedthat g is surjective. See the gure.

28 CHAPTER 6. BIJECTION, INJECTION AND SURJECTION

X1

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Surjective composition: the rst function need not be surjective.

6.3 BijectionMain article: Bijective functionA function is bijective if it is both injective and surjective. A bijective function is a bijection (one-to-one corre-spondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. Thisequivalent condition is formally expressed as follow.

The function f : A! B is bijective i for all b 2 B , there is a unique a 2 A such that f(a) = b:

A function f : A B is bijective if and only if it is invertible, that is, there is a function g: B A such thatg o f = identity function on A and f o g = identity function on B. This function maps each image to its uniquepreimage.

The composition of two bijections is again a bijection, but if g o f is a bijection, then it can only be concludedthat f is injective and g is surjective. (See the gure at right and the remarks above regarding injections andsurjections.)

The bijections from a set to itself form a group under composition, called the symmetric group.

6.4 CardinalitySuppose you want to dene what it means for two sets to have the same number of elements. One way to do this isto say that two sets have the same number of elements if and only if all the elements of one set can be paired withthe elements of the other, in such a way that each element is paired with exactly one element. Accordingly, we candene two sets to have the same number of elements if there is a bijection between them. We say that the two setshave the same cardinality.Likewise, we can say that set A has fewer than or the same number of elements as set B if there is an injectionfrom A to B . We can also say that set A has fewer than the number of elements in set B if there is an injectionfrom A to B but not a bijection between A and B .

6.5. EXAMPLES 29

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Bijective composition: the rst function need not be surjective and the second function need not be injective.

6.5 ExamplesIt is important to specify the domain and codomain of each function since by changing these, functions which wethink of as the same may have dierent jectivity.

6.5.1 Injective and surjective (bijective)

For every set A the identity function idA and thus specically R! R : x 7! x .

R+ ! R+ : x 7! x2 and thus also its inverse R+ ! R+ : x 7! px .

The exponential function exp : R ! R+ : x 7! ex and thus also its inverse the natural logarithm ln : R+ !R : x 7! lnx

6.5.2 Injective and non-surjective

The exponential function exp : R! R : x 7! ex

6.5.3 Non-injective and surjective

R! R : x 7! (x 1)x(x+ 1) = x3 x

R! [1; 1] : x 7! sin(x)

6.5.4 Non-injective and non-surjective

R! R : x 7! sin(x)

30 CHAPTER 6. BIJECTION, INJECTION AND SURJECTION

6.6 Properties For every function f, subsetA of the domain and subsetB of the codomainwe haveA f 1(f(A)) and f(f 1(B)) B. If f is injective we have A = f 1(f(A)) and if f is surjective we have f(f 1(B)) = B.

For every function h : A C we can dene a surjection H : A h(A) : a h(a) and an injection I : h(A) C : a a. It follows that h = I H. This decomposition is unique up to isomorphism.

6.7 Category theoryIn the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms,and isomorphisms, respectively.

6.8 HistoryThis terminology was originally coined by the Bourbaki group.

6.9 See also Bijective function Horizontal line test Injective module Injective function Permutation Surjective function

6.10 External links Earliest Uses of Some of theWords ofMathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.

Chapter 7

Block design

This article is about block designs with xed block size (uniform). For block designs with variable block sizes, seeCombinatorial design. For experimental designs in statistics, see randomized block design.

In combinatorial mathematics, a block design is a set together with a family of subsets (repeated subsets are allowedat times) whose members are chosen to satisfy some set of properties that are deemed useful for a particular appli-cation. These applications come from many areas, including experimental design, nite geometry, software testing,cryptography, and algebraic geometry. Many variations have been examined, but the most intensely studied are thebalanced incomplete block designs (BIBDs or 2-designs) which historically were related to statistical issues in thedesign of experiments.[1][2]

A block design in which all the blocks have the same size is called uniform. The designs discussed in this article areall uniform. Pairwise balanced designs (PBDs) are examples of block designs that are not necessarily uniform.

7.1 Denition of a BIBD (or 2-design)Given a nite set X (of elements called points) and integers k, r, 1, we dene a 2-design (or BIBD, standing forbalanced incomplete block design) B to be a family of k-element subsets of X, called blocks, such that the numberr of blocks containing x in X is not dependent on which x is chosen, and the number of blocks containing givendistinct points x and y in X is also independent of the choices.Family in the above denition can be replaced by set if repeated blocks are not allowed. Designs in which repeatedblocks are not allowed are called simple.Here v (the number of elements of X, called points), b (the number of blocks), k, r, and are the parameters of thedesign. (To avoid degenerate examples, it is also assumed that v > k, so that no block contains all the elements of theset. This is the meaning of incomplete in the name of these designs.) In a table:

The design is called a (v, k, )-design or a (v, b, r, k, )-design. The parameters are not all independent; v, k, and determine b and r, and not all combinations of v, k, and are possible. The two basic equations connecting theseparameters are

bk = vr;

(v 1) = r(k 1):These conditions are not sucient as, for example, a (43,7,1)-design does not exist.[3]

The order of a 2-design is dened to be n = r . The complement of a 2-design is obtained by replacing each blockwith its complement in the point set X. It is also a 2-design and has parameters v = v, b = b, r = b r, k = v k, = + b 2r. A 2-design and its complement have the same order.A fundamental theorem, Fishers inequality, named after the statistician Ronald Fisher, is that b v in any 2-design.

31

32 CHAPTER 7. BLOCK DESIGN

7.2 Symmetric BIBDsThe case of equality in Fishers inequality, that is, a 2-design with an equal number of points and blocks, is calleda symmetric design.[4] Symmetric designs have the smallest number of blocks amongst all the 2-designs with thesame number of points.In a symmetric design r = k holds as well as b = v, and, while it is generally not true in arbitrary 2-designs, in asymmetric design every two distinct blocks meet in points.[5] A theorem of Ryser provides the converse. If X isa v-element set, and B is a v-element set of k-element subsets (the blocks), such that any two distinct blocks haveexactly points in common, then (X, B) is a symmetric block design.[6]

The parameters of a symmetric design satisfy

(v 1) = k(k 1):

This imposes strong restrictions on v, so the number of points is far from arbitrary. The BruckRyserChowlatheorem gives necessary, but not sucient, conditions for the existence of a symmetric design in terms of theseparameters.The following are important examples of symmetric 2-designs:

7.2.1 Projective planesMain article: Projective plane

Finite projective planes are symmetric 2-designs with = 1 and order n > 1. For these designs the symmetric designequation becomes:

v 1 = k(k 1):

Since k = r we can write the order of a projective plane as n = k 1 and, from the displayed equation above, we obtainv = (n + 1)n + 1 = n2 + n + 1 points in a projective plane of order n.As a projective plane is a symmetric design, we have b = v, meaning that b = n2 + n + 1 also. The number b is thenumber of lines of the projective plane. There can be no repeated lines since = 1, so a projective plane is a simple2-design in which the number of lines and the number of points are always the same. For a projective plane, k is thenumber of points on each line and it is equal to n + 1. Similarly, r = n + 1 is the number of lines with which a givenpoint is incident.For n = 2 we get a projective plane of order 2, also called the Fano plane, with v = 4 + 2 + 1 = 7 points and 7 lines.In the Fano plane, each line has n + 1 = 3 points and each point belongs to n + 1 = 3 lines.Projective planes are known to exist for all orders which are prime numbers or powers of primes. They form the onlyknown innite family (with respect to having a constant value) of symmetric block designs.[7]

7.2.2 BiplanesA biplane or biplane geometry is a symmetric 2-design with = 2; that is, every set of two points is contained intwo blocks (lines), while any two lines intersect in two points.[7] They are similar to nite projective planes, exceptthat rather than two points determining one line (and two lines determining one point), two points determine twolines (respectively, points). A biplane of order n is one whose blocks have k = n + 2 points; it has v = 1 + (n + 2)(n +1)/2 points (since r = k).The 18 known examples[8] are listed below.

(Trivial) The order 0 biplane has 2 points (and lines of size 2; a 2-(2,2,2) design); it is two points, with twoblocks, each consisting of both points. Geometrically, it is the digon.

7.3. RESOLVABLE 2-DESIGNS 33

The order 1 biplane has 4 points (and lines of size 3; a 2-(4,3,2) design); it is the complete design with v = 4and k = 3. Geometrically, the points are the vertices and the blocks are the faces of a tetrahedron.

The order 2 biplane is the complement of the Fano plane: it has 7 points (and lines of size 4; a 2-(7,4,2)),where the lines are given as the complements of the (3-point) lines in the Fano plane.[9]

The order 3 biplane has 11 points (and lines of size 5; a 2-(11,5,2)), and is also known as the Paley biplaneafter Raymond Paley; it is associated to the Paley digraph of order 11, which is constructed using the eld with11 elements, and is the Hadamard 2-design associated to the size 12 Hadamard matrix; see Paley constructionI.

Algebraically this corresponds to the exceptional embedding of the projective special linear groupPSL(2,5)in PSL(2,11) see projective linear group: action on p points for details.[10]

There are three biplanes of order 4 (and 16 points, lines of size 6; a 2-(16,6,2)). These three designs are alsoMenon designs.

There are four biplanes of order 7 (and 37 points, lines of size 9; a 2-(37,9,2)).[11]

There are ve biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)).[12]

Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)).[13]

7.2.3 Hadamard 2-designsAn Hadamard matrix of size m is an m m matrix H whose entries are 1 such that HH = mI, where H is thetranspose of H and Im is the m m identity matrix. An Hadamard matrix can be put into standardized form (that is,converted to an equivalent Hadamard matrix) where the rst row and rst column entries are all +1. If the size m >2 then m must be a multiple of 4.Given an Hadamard matrix of size 4a in standardized form, remove the rst row and rst column and convert every1 to a 0. The resulting 01 matrixM is the incidence matrix of a symmetric 2-(4a 1, 2a 1, a 1) design calledan Hadamard 2-design.[14] This construction is reversible, and the incidence matrix of a symmetric 2-design withthese parameters can be used to form an Hadamard matrix of size 4a.

7.3 Resolvable 2-designsA resolvable 2-design is a BIBD whose blocks can be partitioned into sets (called parallel classes), each of whichforms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design.If a 2-(v,k,) resolvable design has c parallel classes, then b v + c 1.[15]

Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution.[16]

Archetypical resolvable 2-designs are the nite ane planes. A solution of the famous 15 schoolgirl problem is aresolution of a 2-(15,3,1) design.[17]

7.4 Generalization: t-designsGiven any positive integer t, a t-design B is a class of k-element subsets of X, called blocks, such that every point x inX appears in exactly r blocks, and every t-element subset T appears in exactly blocks. The numbers v (the numberof elements of X), b (the number of blocks), k, r, , and t are the parameters of the design. The design may be calleda t-(v,k,)-design. Again, these four numbers determine b and r and the four numbers themselves cannot be chosenarbitrarily. The equations are

i =

v it i

k it i

for i = 0; 1; : : : ; t;

34 CHAPTER 7. BLOCK DESIGN

where i is the number of blocks that contain any i-element set of points.Theorem:[18] Any t-(v,k,)-design is also an s-(v,k,)-design for any s with 1 s t. (Note that the lambda valuechanges as above and depends on s.)A consequence of this theorem is that every t-design with t 2 is also a 2-design.There are no known examples of non-trivial t-(v,k,1)-designs with t > 5 .The term block design by itself usually means a 2-design.

7.4.1 Derived and extendable t-designsLet D = (X, B) be a t-(v,k,) design and p a point of X. The derived design Dp has point set X {p} and as block setall the blocks of D which contain p with p removed. It is a (t 1)-(v 1, k 1, ) design. Note that derived designswith respect to dierent points may not be isomorphic. A design E is called an extension of D if E has a point p suchthat E is isomorphic to D; we call D extendable if it has an extension.Theorem:[19] If a t-(v,k,) design has an extension, then k + 1 divides b(v + 1).The only extendable projective planes (symmetric 2-(n2 + n + 1, n + 1, 1) designs) are those of orders 2 and 4.[20]

Every Hadamard 2-design is extendable (to an Hadamard 3-design).[21]

Theorem:.[22] If D, a symmetric 2-(v,k,) design, is extendable, then one of the following holds:

1. D is an Hadamard 2-design,2. v = ( + 2)(2 + 4 + 2), k = 2 + 3 + 1,3. v = 495, k = 39, = 3.

Note that the projective plane of order two is an Hadamard 2-design; the projective plane of order four has parameterswhich fall in case 2; the only other known symmetric 2-designs with parameters in case 2 are the order 9 biplanes,but none of them are extendable; and there is no known symmetric 2-design with the parameters of case 3.[23]

Inversive planes

A design with the parameters of the extension of an ane plane, i.e., a 3-(n2 + 1, n + 1, 1) design, is called a niteinversive plane, or Mbius plane, of order n.It is possible to give a geometric description of some inversive planes, indeed, of all known inversive planes. Anovoid in PG(3,q) is a set of q2 + 1 points, no three collinear. It can be shown that every plane (which is a hyperplanesince the geometric dimension is 3) of PG(3,q) meets an ovoid O in either 1 or q + 1 points. The plane sections ofsize q + 1 of O are the blocks of an inversive plane of order q. Any inversive plane arising this way is called egglike.All known inversive planes are egglike.An example of an ovoid is the elliptic quadric, the set of zeros of the quadratic form

x1x2 + f(x3, x4),

where f is an irreducible quadratic form in two variables over GF(q). [f(x,y) = x2 + xy + y2 for example].If q is an odd power of 2, another type of ovoid is known the SuzukiTits ovoid.Theorem. Let q be a positive integer, at least 2. (a) If q is odd, then any ovoid is projectively equivalent to the ellipticquadric in a projective geometry PG(3,q); so q is a prime power and there is a unique egglike inversive plane of orderq. (But it is unknown if non-egglike ones exist.) (b) if q is even, then q is a power of 2 and any inversive plane oforder q is egglike (but there may be some unknown ovoids).

7.5 Steiner systemsMain article: Steiner system

7.6. PARTIALLY BALANCED DESIGNS (PBIBDS) 35

A Steiner system (named after Jakob Steiner) is a t-design with = 1 and t 2.A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsetsof S (called blocks) with the property that each t-element subset of S is contained in exactly one block. In the generalnotation for block designs, an S(t,k,n) would be a t-(n,k,1) design.This denition is relatively modern, generalizing the classical denition of Steiner systems which in addition requiredthat k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple system, while an S(3,4,n) was called a Steinerquadruple system, and so on. With the generalization of the denition, this naming system is no longer strictlyadhered to.Projective planes and ane planes are examples of Steiner systems under the current denition while only the Fanoplane (projective plane of order 2) would have been a Steiner system under the older denition.

7.6 Partially balanced designs (PBIBDs)An n-class association scheme consists of a set X of size v together with a partition S of X X into n + 1 binaryrelations, R0, R1, ..., R. A pair of elements in relation R are said to be ithassociates. Each element of X has n ithassociates. Furthermore:

R0 = f(x; x) : x 2 Xg and is called the Identity relation. Dening R := f(x; y)j(y; x) 2 Rg , if R in S, then R* in S If (x; y) 2 Rk , the number of z 2 X such that (x; z) 2 Ri and (z; y) 2 Rj is a constant pkij depending on i,j, k but not on the particular choice of x and y.

An association scheme is commutative if pkij = pkji for all i, j and k. Most authors assume this property.A partially balanced incomplete block design with n associate classes (PBIBD(n)) is a block design based on av-set X with b blocks each of size k and with each element appearing in r blocks, such that there is an associationscheme with n classes dened on X where, if elements x and y are ith associates, 1 i n, then they are together inprecisely blocks.A PBIBD(n) determines an association scheme but the converse is false.[24]

7.6.1 ExampleLet A(3) be the following association scheme with three associate classes on the set X = {1,2,3,4,5,6}. The (i,j) entryis s if elements i and j are in relation R.

The blocks of a PBIBD(3) based on A(3) are:

The parameters of this PBIBD(3) are: v = 6, b = 8, k = 3, r = 4 and 1 = 2 = 2 and 3 = 1. Also, for the associationscheme we have n0 = n2 = 1 and n1 = n3 = 2.[25]

7.6.2 PropertiesThe parameters of a PBIBD(m) satisfy:[26]

1. vr = bk

2. Pmi=1 ni = v 1

36 CHAPTER 7. BLOCK DESIGN

3. Pmi=1 nii = r(k 1)4. Pmu=0 phju = nj5. nipijh = njpjih

A PBIBD(1) is a BIBD and a PBIBD(2) in which 1 = 2 is a BIBD.[27]

7.6.3 Two associate class PBIBDsPBIBD(2)s have been studied the most since they are the simplest and most useful of the PBIBDs.[28] They fall intosix types[29] based on a classication of the then known PBIBD(2)s by Bose & Shimamoto (1952):[30]

1. group divisible;

2. triangular;

3. Latin square type;

4. cyclic;

5. partial geometry type;

6. miscellaneous.

7.7 ApplicationsThe mathematical subject of block designs originated in the statistical framework of design of experiments. Thesedesigns were especially useful in applications of the technique of analysis of variance (ANOVA). This remains asignicant area for the use of block designs.While the origins of the subject are grounded in biological applications (as is some of the existing terminology), thedesigns are used in many applications where systematic comparisons are being made, such as in software testing.The incidence matrix of block designs provide a natural source of interesting block codes that are used as errorcorrecting codes. The rows of their incidence matrices are also used as the symbols in a form of pulse-positionmodulation.[31]

7.8 See also Incidence geometry

7.9 Notes[1] Colbourn & Dinitz 2007

[2] Stinson 2003, pg.1

[3] Proved by Tarry in 1900 who showed that there was no pair of orthogonal Latin squares of order six. The 2-design withthe indicate parameters is equivalent to the existence of ve mutually orthogonal Latin squares of order six.

[4] They have also been referred to as projective designs or square designs. These alternatives have been used in an attempt toreplace the term symmetric, since there is nothing symmetric (in the usual meaning of the term) about these designs. Theuse of projective is due to P.Dembowski (Finite Geometries, Springer, 1968), in analogy with the most common example,projective planes, while square is due to P. Cameron (Designs, Graphs, Codes and their Links, Cambridge, 1991) andcaptures the implication of v = b on the incidence matrix. Neither term has caught on as a replacement and these designsare still universally referred to as symmetric.

7.10. REFERENCES 37

[5] Stinson 2003, pg.23, Theorem 2.2

[6] Ryser 1963, pp. 102104

[7] Hughes & Piper 1985, pg.109

[8] Hall 1986, pp.320-335

[9] Assmus & Key 1992, pg.55

[10] Martin, Pablo; Singerman, David (April 17, 2008), From Biplanes to the Klein quartic and the Buckyball (PDF), p. 4

[11] Salwach & Mezzaroba 1978

[12] Kaski & stergrd 2008

[13] Aschbacher 1971, pp. 279281

[14] Stinson 2003, pg. 74, Theorem 4.5

[15] Hughes & Piper 1985, pg. 156, Theorem 5.4

[16] Hughes & Piper 1985, pg. 158, Corollary 5.5

[17] Beth, Jungnickel & Lenz 1986, pg. 40 Example 5.8

[18] Stinson 2003, pg.203, Corollary 9.6

[19] Hughes & Piper 1985, pg.29

[20] Cameron & van Lint 1991, pg. 11, Proposition 1.34

[21] Hughes & Piper 1985, pg. 132, Theorem 4.5

[22] Cameron & van Lint 1991, pg. 11, Theorem 1.35

[23] Colbourn & Dinitz 2007, pg. 114, Remarks 6.35

[24] Street & Street 1987, pg. 237

[25] Street & Street 1987, pg. 238

[26] Street & Street 1987, pg. 240, Lemma 4

[27] Colburn & Dinitz 2007, pg. 562, Remark 42.3 (4)

[28] Street & Street 1987, pg. 242

[29] Not a mathematical classication since one of the types is a catch-all and everything else.

[30] Raghavarao 1988, pg. 127

[31] Noshad, Mohammad; Brandt-Pearce, Maite (Jul 2012). Expurgated PPM Using Symmetric Balanced Incomplete BlockDesigns. IEEE Communications Letters 16 (7): 968971. doi:10.1109/LCOMM.2012.042512.120457.

7.10 References Aschbacher, Michael (1971). On collineation groups of symmetric block designs. Journal of CombinatorialTheory, Series A 11 (3): 272281. doi:10.1016/0097-3165(71)90054-9.

Assmus, E.F.; Key, J.D. (1992), Designs and Their Codes, Cambridge: Cambridge University Press, ISBN0-521-41361-3

Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge UniversityPress. 2nd ed. (1999) ISBN 978-0-521-44432-3.

R. C. Bose, A Note on Fishers Inequality for Balanced Incomplete Block Designs, Annals of MathematicalStatistics, 1949, pages 619620.

38 CHAPTER 7. BLOCK DESIGN

Bose, R. C.; Shimamoto, T. (1952), Classication and analysis of partially balanced incomplete block designswith two associate classes, Journal of the American Statistical Association 47: 151184, doi:10.1080/01621459.1952.10501161

Cameron, P. J.; van Lint, J. H. (1991), Designs, Graphs, Codes and their Links, Cambridge: Cambridge Uni-versity Press, ISBN 0-521-42385-6

Colbourn, Charles J.; Dinitz, Jerey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:Chapman & Hall/ CRC, ISBN 1-58488-506-8

R. A. Fisher, An examination of the dierent possible solutions of a problem in incomplete blocks, Annalsof Eugenics, volume 10, 1940, pages 5275.

Hall, Jr., Marshall (1986), Combinatorial Theory (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-09138-3

Hughes, D.R.; Piper, E.C. (1985), Design theory, Cambridge: Cambridge University Press, ISBN 0-521-25754-9

Kaski, Petteri and stergrd, Patric (2008). There Are Exactly Five Biplanes with k = 11. Journal ofCombinatorial Designs 16 (2): 117127. doi:10.1002/jcd.20145. MR 2008m:05038.

Lander, E. S. (1983), Symmetric Designs: An Algebraic Approach, Cambridge: Cambridge University Press

Lindner, C.C.; Rodger, C.A. (1997), Design Theory, Boca Raton: CRC Press, ISBN 0-8493-3986-3

Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (correctedreprint of the 1971 Wiley ed.). New York: Dover.

Raghavarao, Damaraju and Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications.World Scientic.

Ryser, Herbert John (1963), Chapter 8: Combinatorial Designs, Combinatorial Mathematics (Carus Mono-graph #14), Mathematical Association of America

Salwach, Chester J.; Mezzaroba, Joseph A. (1978). The four biplanes with k = 9. Journal of CombinatorialTheory, Series A 24 (2): 141145. doi:10.1016/0097-3165(78)90002-X.

S. S. Shrikhande, and Vasanti N. Bhat-Nayak, Non-isomorphic solutions of some balanced incomplete blockdesigns I Journal of Combinatorial Theory, 1970

Stinson, Douglas R. (2003), Combinatorial Designs: Constructions and Analysis, New York: Springer, ISBN0-387-95487-2

Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P.[Clarendon]. pp. 400+xiv. ISBN 0-19-853256-3.

van Lint, J.H.; Wilson, R.M. (1992). A Course in Combinatorics. Cambridge: Cambridge University Press.

7.11 External links DesignTheory.Org: Databases of combinatorial, statistical, and experimental block designs. Software andother resources hosted by the School of Mathematical Sciences at Queen Mary College, University of London.

Design Theory Resources: Peter Cameron's page of web based design theory resources.

Weisstein, Eric W., Block Designs, MathWorld.

Chapter 8

Carathodorys theorem (convex hull)

(0,1)

(0,0) (1,0)

(1,1)

(1/4,1/4)

An illustration of Carathodorys theorem for a square in R2

See also Carathodorys theorem (disambiguation) for other meanings

In convex geometry Carathodorys theorem states that if a point x of Rd lies in the convex hull of a set P, there

39

40 CHAPTER 8. CARATHODORYS THEOREM (CONVEX HULL)

is a subset P of P consisting of d + 1 or fewer points such that x lies in the convex hull of P. Equivalently, x liesin an r-simplex with vertices in P, where r d . The result is named for Constantin Carathodory, who proved thetheorem in 1911 for the case when P is compact. In 1914 Ernst Steinitz expanded Carathodorys theorem for anysets P in Rd.For example, consider a set P = {(0,0), (0,1), (1,0), (1,1)} which is a subset of R2. The convex hull of this setis a square. Consider now a point x = (1/4, 1/4), which is in the convex hull of P. We can then construct a set{(0,0),(0,1),(1,0)} = P, the convex hull of which is a triangle and encloses x, and thus the theorem works for thisinstance, since |P| = 3. It may help to visualise Carathodorys theorem in 2 dimensions, as saying that we canconstruct a triangle consisting of points from P that encloses any point in P.

8.1 ProofLet x be a point in the convex hull of P. Then, x is a convex combination of a nite number of points in P :

x =kX

j=1

jxj

where every x is in P, every is non-negative, andPkj=1 j = 1 .Suppose k > d + 1 (otherwise, there is nothing to prove). Then, the points x2 x1, ..., xk x1 are linearly dependent,so there are real scalars 2, ..., k, not all zero, such that

kXj=2

j(xj x1) = 0:

If 1 is dened as

1 := kX

j=2

j

then

kXj=1

jxj = 0

kXj=1

j = 0

and not all of the j are equal to zero. Therefore, at least one > 0. Then,

x =kX

j=1

jxj kX

j=1

jxj =kX

j=1

(j j)xj

for any real . In particular, the equality will hold if is dened as

:= min1jk

njj

: j > 0o= ii :

Note that > 0, and for every j between 1 and k,

8.2. SEE ALSO 41

j j 0:

In particular, i i = 0 by denition of . Therefore,

x =kX

j=1

(j j)xj

where every j j is nonnegative, their sum is one , and furthermore, i i = 0 . In other words, x isrepresented as a convex combination of at most k1 points of P. This process can be repeated until x is representedas a convex combination of at most d + 1 points in P.An alternative proof uses Hellys theorem.

8.2 See also ShapleyFolkman lemma Hellys theorem KreinMilman theorem Choquet theory

8.3 References Carathodory, C. (1911), "ber den Variabilittsbereich der Fourierschen Konstanten von positiven harmonis-chen Funktionen, Rendiconti del Circolo Matematico di Palermo 32: 193217, doi:10.1007/BF03014795.

Danzer, L.; Grnbaum, B.; Klee, V. (1963), Hellys theorem and its relatives, Convexity, Proc. Symp. PureMath. 7, American Mathematical Society, pp. 101179.

Eckho, J. (1993), Helly, Radon, and Carathodory type theorems, Handbook of Convex Geometry A, B,Amsterdam: North-Holland, pp. 389448.

Steinitz, Ernst (1913), Bedingt konvergente Reihen und konvexe Systeme, J. Reine Angew. Math. 143 (143):128175, doi:10.1515/crll.1913.143.128.

8.4 External links Concise statement of theorem in terms of convex hulls (at PlanetMath)

Chapter 9

Cartesian product

Cartesian square redirects here. For Cartesian squares in category theory, see Cartesian square (category theory).In mathematics, a Cartesian product is a mathematical operation which returns a set (or product set or simply

(z,1) (z,2) (z,3)

(y,1) (y,2) (y,3)

(x,1) (x,2) (x,3)

1 2 3

z

y

x

BA

AB

Cartesian product AB of the sets A=fx;y;zg and B=f1;2;3g

product) from multiple sets. That is, for sets A and B, t