Lipschutz s , lipson m schaums outline of discrete maths

Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Front Matter © The McGraw−Hill Companies, 2004 Preface Preface Discrete mathematics, the study of ®nite systems, has become increasingly important as the computer age has advanced The digital computer is basically a ®nite structure, and many of its properties can be understood and interpreted within the framework of ®nite mathematical systems This book, in presenting the more essential material, may be used as a textbook for a formal course in discrete mathematics or as a supplement to all current texts The ®rst three chapters cover the standard material on sets, relations, and functions and algorithms Next come chapters on logic, vectors and matrices, counting, and probability We than have three chapters on graph theory: graphs, directed graphs, and binary trees Finally there are individual chapters on properties of the integers, algebraic systems, languages and machines, ordered sets and lattices, and Boolean algebra The chapter on functions and algorithms includes a discussion of cardinality and countable sets, and complexity The chapters on graph theory include discussions on planarity, traversability, minimal paths, and Warshall's and Hu€man's algorithms The chapter on languages and machines includes regular expressions, automata, and Turing machines and computable functions We emphasize that the chapters have been written so that the order can be changed without diculty and without loss of continuity This second edition of Discrete Mathmatics covers much more material and in greater depth than the ®rst edition The topics of probability, regular expressions and regular sets, binary trees, cardinality, complexity, and Turing machines and computable functions did not appear in the ®rst edition or were only mentioned This new material re¯ects the fact that discrete mathematics now is mainly a one-year course rather than a one-semester course Each chapter begins with a clear statement of pertinent de®nition, principles, and theorems with illustrative and other descriptive material This is followed by sets of solved and supplementary problems The solved problems serve to illustrate and amplify the material, and also include proofs of theorems The supplementary problems furnish a complete review of the material in the chapter More material has been included than can be covered in most ®rst courses This has been done to make the book more ¯exible, to provide a more useful book of reference, and to stimulate further interest in the topics Finally, we wish to thank the sta€ of the McGraw-Hill Schaum's Outline Series, especially Arthur Biderman and Maureen Walker, for their unfailing cooperation SEYMOUR LIPSCHUTZ MARC LARS LIPSON v Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Set Theory © The McGraw−Hill Companies, 2004 Text Chapter Set Theory 1.1 INTRODUCTION The concept of a set appears in all mathematics This chapter introduces the notation and terminology of set theory which is basic and used throughout the text Though logic is formally treated in Chapter 4, we introduce Venn diagram representation of sets here, and we show how it can be applied to logical arguments The relation between set theory and logic will be further explored when we discuss Boolean algebra in Chapter 15 This chapter closes with the formal de®nition of mathematical induction, with examples 1.2 SETS AND ELEMENTS A set may be viewed as a collection of objects, the elements or members of the set We ordinarily use capital letters, A, B, X, Y, , to denote sets, and lowercase letters, a, b, x, y, , to denote elements of sets The statement ``p is an element of A'', or, equivalently, ``p belongs to A'', is written pPA The statement that p is not an element of A, that is, the negation of p P A, is written pP = A The fact that a set is completely determined when its members are speci®ed is formally stated as the principle of extension Principle of Extension: Two sets A and B are equal if and only if they have the same members As usual, we write A ˆ B if the sets A and B are equal, and we write A Tˆ B if the sets are not equal Specifying Sets There are essentially two ways to specify a particular set One way, if possible, is to list its members For example, A ˆ fa; e; i; o; ug denotes the set A whose elements are the letters a, e, i, o, u Note that the elements are separated by commas and enclosed in braces { } The second way is to state those properties which characterized the elements in the set For example, B ˆ fx: x is an even integer, x > 0g which reads ``B is the set of x such that x is an even integer and x is greater than 0'', denotes the set B whose elements are the positive integers A letter, usually x, is used to denote a typical member of the set; the colon is read as ``such that'' and the comma as ``and'' EXAMPLE 1.1 (a) The set A above can also be written as A ˆ fx: x is a letter in the English alphabet, x is a vowelg Observe that b P = A, e P A, and p P = A (b) We could not list all the elements of the above set B although frequently we specify the set by writing B ˆ f2; 4; 6; g Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Set Theory © The McGraw−Hill Companies, 2004 Text SET THEORY [CHAP where we assume that everyone knows what we mean Observe that P B but À7 P = B (c) Let E ˆ fx: x2 À 3x ‡ ˆ 0g In other words, E consists of those numbers which are solutions of the equation x2 À 3x ‡ ˆ 0, sometimes called the solution set of the given equation Since the solutions of the equation are and 2, we could also write E ˆ f1; 2g (d ) Let E ˆ fx: x2 À 3x ‡ ˆ 0g, F ˆ f2; 1g and G ˆ f1; 2; 2; 1; 63g Then E ˆ F ˆ G Observe that a set does not depend on the way in which its elements are displayed A set remains the same if its elements are repeated or rearranged Some sets will occur very often in the text and so we use special symbols for them Unless otherwise speci®ed, we will let N Z Q R C ˆ ˆ ˆ ˆ ˆ the the the the the set set set set set of of of of of positive integers: 1, 2, 3, integers: , À2, À1, 0, 1, 2, rational numbers real numbers complex numbers Even if we can list the elements of a set, it may not be practical to so For example, we would not list the members of the set of people born in the world during the year 1976 although theoretically it is possible to compile such a list That is, we describe a set by listing its elements only if the set contains a few elements; otherwise we describe a set by the property which characterizes its elements The fact that we can describe a set in terms of a property is formally stated as the principle of abstraction Principle of Abstraction: Given any set U and any property P, there is a set A such that the elements of A are exactly those members of U which have the property P 1.3 UNIVERSAL SET AND EMPTY SET In any application of the theory of sets, the members of all sets under investigation usually belong to some ®xed large set called the universal set For example, in plane geometry, the universal set consists of all the points in the plane, and in human population studies the universal set consists of all the people in the world We will let the symbol U denote the universal set unless otherwise stated or implied For a given set U and a property P, there may not be any elements of U which have property P For example, the set S ˆ fx: x is a positive integer, x2 ˆ 3g has no elements since no positive integer has the required property The set with no elements is called the empty set or null set and is denoted by D There is only one empty set That is, if S and T are both empty, then S ˆ T since they have exactly the same elements, namely, none 1.4 SUBSETS If every element in a set A is also an element of a set B, then A is called a subset of B We also say that A is contained in B or that B contains A This relationship is written AB or BA Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Set Theory CHAP 1] © The McGraw−Hill Companies, 2004 Text SET THEORY If A is not a subset of B, i.e., if at least one element of A does not belong to B, we write A \ B or B ] A EXAMPLE 1.2 (a) Consider the sets A ˆ f1; 3; 4; 5; 8; 9g B ˆ f1; 2; 3; 5; 7g C ˆ f1; 5g Then C  A and C  B since and 5, the elements of C, are also members of A and B But B \ A since some of its elements, e.g., and 7, not belong to A Furthermore, since the elements of A, B, and C must also belong to the universal set U, we have that U must at least contain the set f1; 2; 3; 4; 5; 6; 7; 8; 9g (b) Let N, Z, Q, and R be de®ned as in Section 1.2 Then NZQR (c) The set E ˆ f2; 4; 6g is a subset of the set F ˆ f6; 2; 4g, since each number 2, 4, and belonging to E also belongs to F In fact, E ˆ F In a similar manner it can be shown that every set is a subset of itself The following properties of sets should be noted: (i) Every set A is a subset of the universal set U since, by de®nition, all the elements of A belong to U Also the empty set D is a subset of A (ii) Every set A is a subset of itself since, trivially, the elements of A belong to A (iii) If every element of A belongs to a set B, and every element of B belongs to a set C, then clearly every element of A belongs to C In other words, if A  B and B  C, then A  C (iv) If A  B and B  A, then A and B have the same elements, i.e., A ˆ B Conversely, if A ˆ B then A  B and B  A since every set is a subset of itself We state these results formally Theorem 1.1: (i) For any set A, we have D  A  U (ii) For any set A, we have A  A (iii) If A  B and B  C, then A  C (iv) A ˆ B if and only if A  B and B  A If A  B, then it is still possible that A ˆ B When A  B but A Tˆ B, we say A is a proper subset of B We will write A & B when A is a proper subset of B For example, suppose A ˆ f1; 3g B ˆ f1; 2; 3g; C ˆ f1; 3; 2g Then A and B are both subsets of C; but A is a proper subset of C, whereas B is not a proper subset of C since B ˆ C 1.5 VENN DIAGRAMS A Venn diagram is a pictoral representation of sets in which sets are represented by enclosed areas in the plane The universal set U is represented by the interior of a rectangle, and the other sets are represented by disks lying within the rectangle If A  B, then the disk representing A will be entirely within the disk representing B as in Fig 1-1(a) If A and B are disjoint, i.e., if they have no elements in common, then the disk representing A will be separated from the disk representing B as in Fig 1-1(b) However, if A and B are two arbitrary sets, it is possible that some objects are in A but not in B, some are in B but not in A, some are in both A and B, and some are in neither A nor B; hence in general we represent A and B as in Fig 1-1(c) Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Set Theory © The McGraw−Hill Companies, 2004 Text SET THEORY [CHAP Fig 1-1 Arguments and Venn Diagrams Many verbal statements are essentially statements about sets and can therefore be described by Venn diagrams Hence Venn diagrams can sometimes be used to determine whether or not an argument is valid Consider the following example EXAMPLE 1.3 Show that the following argument (adapted from a book on logic by Lewis Carroll, the author of Alice in Wonderland) is valid: S1 : S2 : S3 : My saucepans are the only things I have that are made of tin I ®nd all your presents very useful None of my saucepans is of the slightest use S: Your presents to me are not made of tin (The statements S1 , S2 , and S3 above the horizontal line denote the assumptions, and the statement S below the line denotes the conclusion The argument is valid if the conclusion S follows logically from the assumptions S1 , S2 , and S3 ) By S1 the tin objects are contained in the set of saucepans and by S3 the set of saucepans and the set of useful things are disjoint: hence draw the Venn diagram of Fig 1-2 Fig 1-2 Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Set Theory CHAP 1] © The McGraw−Hill Companies, 2004 Text SET THEORY By S2 the set of ``your presents'' is a subset of the set of useful things; hence draw Fig 1-3 Fig 1-3 The conclusion is clearly valid by the above Venn diagram because the set of ``your presents'' is disjoint from the set of tin objects 1.6 SET OPERATIONS This section introduces a number of important operations on sets Union and Intersection The union of two sets A and B, denoted by A ‘ B, is the set of all elements which belong to A or to B; that is, A ‘ B ˆ fx: x P A or x P Bg Here ``or'' is used in the sense of and/or Figure 1-4(a) is a Venn diagram in which A ‘ B is shaded The intersection of two sets A and B, denoted by A ’ B, is the set of elements which belong to both A and B; that is, A ’ B ˆ fx: x P A and x P Bg Figure 1-4(b) is a Venn diagram in which A ’ B is shaded If A ’ B ˆ D, that is, if A and B not have any elements in common, then A and B are said to be disjoint or nonintersecting Fig 1-4 EXAMPLE 1.4 (a) Let A ˆ f1; 2; 3; 4g, B ˆ f3; 4; 5; 6; 7g, C ˆ f2; 3; 5; 7g Then A ‘ B ˆ f1; 2; 3; 4; 5; 6; 7g A ‘ C ˆ f1; 2; 3; 4; 5; 7g A ’ B ˆ f3; 4g A ’ C ˆ f2; 3g Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Set Theory (b) © The McGraw−Hill Companies, 2004 Text SET THEORY [CHAP Let M denote the set of male students in a university C, and let F denote the set of female students in C Then M‘F ˆC since each student in C belongs to either M or F On the other hand, M’F ˆD since no student belongs to both M and F The operation of set inclusion is closely related to the operations of union and intersection, as shown by the following theorem Theorem 1.2: The following are equivalent: A  B, A ’ B ˆ A, and A ‘ B ˆ B Note: This theorem is proved in Problem 1.27 Other conditions equivalent to A  B are given in Problem 1.37 Complements Recall that all sets under consideration at a particular time are subsets of a ®xed universal set U The absolute complement or, simply, complement of a set A, denoted by Ac , is the set of elements which belong to U but which not belong to A; that is, = Ag Ac ˆ fx: x P U, x P  Figure 1-5(a) is a Venn diagram in which Ac is Some texts denote the complement of A by A H or A shaded The relative complement of a set B with respect to a set A or, simply, the di€erence of A and B, denoted by AnB, is the set of elements which belong to A but which not belong to B; that is AnB ˆ fx: x P A; x P = Bg The set AnB is read ``A minus B'' Many texts denote AnB by A À B or A $ B Figure 1-5(b) is a Venn diagram in which AnB is shaded Fig 1-5 EXAMPLE 1.5 Suppose U ˆ N ˆ f1; 2; 3; g, the positive integers, is the universal set Let A ˆ f1; 2; 3; 4; g; B ˆ f3; 4; 5; 6; 7g; C ˆ f6; 7; 8; 9g and let E ˆ f2; 4; 6; 8; g, the even integers Then Ac ˆ f5; 6; 7; 8; g; B c ˆ f1; 2; 8; 9; 10; g; C c ˆ f1; 2; 3; 4; 5; 10; 11; g and AnB ˆ f1; 2g; c BnC ˆ f3; 4; 5g; Also, E ˆ f1; 3; 5; g, the odd integers BnA ˆ f5; 6; 7g; CnE ˆ f7; 9g Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Set Theory © The McGraw−Hill Companies, 2004 Text CHAP 1] SET THEORY Fundamental Products Consider n distinct sets A1 ; A2 ; ; An A fundamental product of the sets is a set of the form AÃ1 ’ AÃ2 ’ Á Á Á ’ AÃn where AÃi is either Ai or Aci We note that (1) there are 2n such fundamental products, (2) any two such fundamental products are disjoint, and (3) the universal set U is the union of all the fundamental products (Problem 1.64) There is a geometrical description of these sets which is illustrated below EXAMPLE 1.6 sets: Consider three sets A, B, and C The following lists the eight fundamental products of the three P1 ˆ A ’ B ’ C; P2 ˆ A ’ B ’ C c P3 ˆ A ’ B c ’ C; P4 ˆ A ’ B c ’ C c ; P5 ˆ Ac ’ B ’ C; P6 ˆ Ac ’ B ’ C c P7 ˆ Ac ’ B c ’ C P8 ˆ Ac ’ B c ’ C c These eight products correspond precisely to the eight disjoint regions in the Venn diagram of sets A, B, C in Fig 1-6 as indicated by the labeling of the regions Fig 1-6 Fig 1-7 Symmetric Di€erence The symmetric di€erence of sets A and B, denoted by A È B, consists of those elements which belong to A or B but not to both; that is, A È B ˆ …A ‘ B†n…A ’ B† One can also show (Problem 1.18) that A È B ˆ …AnB† ‘ …BnA† For example, suppose A ˆ f1; 2; 3; 4; 5; 6g and B ˆ f4; 5; 6; 7; 8; 9g Then AnB ˆ f1; 2; 3g; BnA ˆ f7; 8; 9g and so A È B ˆ f1; 2; 3; 7; 8; 9g Figure 1-7 is a Venn diagram in which A È B is shaded 1.7 ALGEBRA OF SETS AND DUALITY Sets under the operations of union, intersection, and complement satisfy various laws or identities which are listed in Table 1-1 In fact, we formally state this: Theorem 1.3: Sets satisfy the laws in Table 1-1 There are two methods of proving equations involving set operations One way is to use what it means for an object x to be an element of each side, and the other way is to use Venn diagrams For example, consider the ®rst of DeMorgan's laws …A ‘ B†c ˆ Ac ’ B c Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Set Theory © The McGraw−Hill Companies, 2004 Text SET THEORY Table 1-1 [CHAP Laws of the algebra of sets Idempotent laws (1a) A‘AˆA (1b) A’AˆA Associative laws (2a) …A ‘ B† ‘ C ˆ A ‘ …B ‘ C† (3a) A‘BˆB‘A (2b) …A ’ B† ’ C ˆ A ’ …B ’ C† Commutative laws (3b) A’BˆB’A Distributive laws (4a) A ‘ …B ’ C† ˆ …A ‘ B† ’ …A ‘ C† (4b) A ’ …B ‘ C† ˆ …A ’ B† ‘ …A ’ C† Identity laws (5a) A‘DˆA (6a) A‘U ˆU (5b) A’U ˆA (6b) A’DˆD Involution laws (7) …Ac †c ˆ A Complement laws (8a) A ‘ Ac ˆ U (8b) A ’ Ac ˆ D (9a) Uc ˆD (9b) Dc ˆ U DeMorgan's laws (10a) …A ‘ B†c ˆ Ac ’ B c (10b) …A ’ B†c ˆ Ac ‘ B c Method 1: We ®rst show that …A ‘ B†c  Ac ’ B c If x P …A ‘ B†c , then x P = A ‘ B Thus x P = A and x P = B, and so x P Ac and x P B c Hence x P Ac ’ B c Next we show that Ac ’ B c  …A ‘ B†C Let x P Ac ’ B c Then x P Ac and x P B c , so x P = A and x P = B Hence x P = A ‘ B, so x P …A ‘ B†c We have proven that every element of …A ‘ B†c belongs to Ac ’ Bc and that every element of Ac ’ Bc belongs to …A ‘ B†c Together, these inclusions prove that the sets have the same elements, i.e., that …A ‘ B†c ˆ Ac ’ Bc Method 2: From the Venn diagram for A ‘ B in Fig 1-4, we see that …A ‘ B†c is represented by the shaded area in Fig 1-8(a) To ®nd Ac ’ B c , the area in both Ac and B c , we shaded Ac with strokes in one direction and B c with strokes in another direction as in Fig 1-8(b) Then Ac ’ B c is represented by the crosshatched area, which is shaded in Fig 1-8(c) Since …A ‘ B†c and Ac ’ B c are represented by the same area, they are equal Fig 1-8 Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Set Theory CHAP 1] © The McGraw−Hill Companies, 2004 Text SET THEORY Duality Note that the identities in Table 1-1 are arranged in pairs, as, for example, (2a) and (2b) We now consider the principle behind this arrangement Suppose E is an equation of set algebra The dual E à of E is the equation obtained by replacing each occurrence of ‘, ’, U, and D in E by ’, ‘, D and U, respectively For example, the dual of …U ’ A† ‘ …B ’ A† ˆ A is …D ‘ A† ’ …B ‘ A† ˆ A Observe that the pairs of laws in Table 1-1 are duals of each other It is a fact of set algebra, called the principle of duality, that, if any equation E is an identity, then its dual E à is also an identity 1.8 FINITE SETS, COUNTING PRINCIPLE A set is said to be ®nite if it contains exactly m distinct elements where m denotes some nonnegative integer Otherwise, a set is said to be in®nite For example, the empty set D and the set of letters of the English alphabet are ®nite sets, whereas the set of even positive integers, f2; 4; 6; g, is in®nite The notation n…A† will denote the number of elements in a ®nite set A Some texts use #…A†; jAj or card…A† instead of n…A† Lemma 1.4: If A and B are disjoint ®nite sets, then A ‘ B is ®nite and n…A ‘ B† ˆ n…A† ‡ n…B† Proof In counting the elements of A ‘ B, ®rst count those that are in A There are n…A† of these The only other elements of A ‘ B are those that are in B but not in A But since A and B are disjoint, no element of B is in A, so there are n…B† elements that are in B but not in A Therefore, n…A ‘ B† ˆ n…A† ‡ n…B† We also have a formula for n…A ‘ B† even when they are not disjoint This is proved in Problem 1.28 Theorem 1.5: If A and B are ®nite sets, then A ‘ B and A ’ B are ®nite and n…A ‘ B† ˆ n…A† ‡ n…B† À n…A ’ B† We can apply this result to obtain a similar formula for three sets: Corollary 1.6: If A, B, and C are ®nite sets, then so is A ‘ B ‘ C, and n…A ‘ B ‘ C† ˆ n…A† ‡ n…B† ‡ n…C† À n…A ’ B† À n…A ’ C† À n…B ’ C† ‡ n…A ’ B ’ C† Mathematical induction (Section 1.10) may be used to further generalize this result to any ®nite number of sets EXAMPLE 1.7 Consider the following data for 120 mathematics students at a college concerning the languages French, German, and Russian: 65 study French 45 study German 42 study Russian 20 study French and German 25 study French and Russian 15 study German and Russian study all three languages Let F, G, and R denote the sets of students studying French, German and Russian, respectively We wish to ®nd the number of students who study at least one of the three languages, and to ®ll in the correct number of students in each of the eight regions of the Venn diagram shown in Fig 1-9 Fig 1-9 ˆ bs1 Bab; (b) ˆ s2 BaBab; ( f ) ˆ  ˆ s2 aBab: (c) ˆ s2 bBab; 13.80  ˆ s0 a; q2 ˆ s1 Bas0 N: ˆ s1 b, q1 ˆ s0 abs1 N; 13.81 Yes 13.82 No, since  3  3 Á Á Á never ends 13.83 q1 q3 q5 q8 ˆ s0 BBsN R (NO); q2 ˆ s0 bbsN R (NO); ˆ s0 aas1 R; q4 ˆ s1 BBsN R (NO); ˆ s1 aasN R (NO); q6 ˆ s1 bbs2 R; q7 ˆ s2 bbs2 R; ˆ s2 aasN R (NO); q9 ˆ s2 BBsY R (YES) 13.84 q1 q3 q5 q7 q9 ˆ s0 BBsN R (NO); q2 ˆ s0 bbsN R (NO); ˆ s0 aas1 R; q4 ˆ s1 BBsY R (accepts); ˆ s1 bbsN R (NO); q6 ˆ s1 aas2 R; ˆ s2 BBsY R (YES); q8 ˆ s2 aasN R (NO); ˆ s2 bbsN N (NO) (d )  ˆ s0 1111B111: (e)  is not changed by q; (d ) ˆ s3 BbBab; (e)  is not changed by q; Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 13 Languages, Grammars, Machines CHAP 13] LANGUAGES, GRAMMARS, MACHINES 13.85 (a) h6i ˆ 17 ; (b) hmi ˆ 16 B1B14 B12 ; 13.86 (a) ‰EŠ ˆ 7; © The McGraw−Hill Companies, 2004 Text (b) ‰EŠ ˆ 2; (c) hmi ˆ 1B1B1; (c) ‰EŠ ˆ 14: 13.87 Strategy: Erase ®rst three 1s: q2 ˆ s1 BBsH N (halt), q1 ˆ s0 1Bs1 R, q5 ˆ s2 1BsH N q4 ˆ s2 BBsH N (halt), q3 ˆ s1 1Bs2 R, 13.88 Strategy: Erase ®rst and then all 1s after B: q2 ˆ s1 11s1 R, q3 ˆ s1 BBs2 R; q1 ˆ s0 1Bs1 R, q4 ˆ s2 1Bs3 R; q5 ˆ s3 1Bs3 R, q6 ˆ s3 BBsH N (halt) (d ) not de®ned 441 Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 14 Ordered Sets and Lattices © The McGraw−Hill Companies, 2004 Text Chapter 14 Ordered Sets and Lattices 14.1 INTRODUCTION Order and precedence relationships appear in many di€erent places in mathematics and computer science This chapter makes these notions precise We also de®ne a lattice, which is a special kind of an ordered set 14.2 ORDERED SETS Suppose R is a relation on a set S satisfying the following three properties: ‰O1 Š (Re¯exive) For any a P S, we have a R a: ‰O2 Š (Antisymmetric) If a R b and b R a, then a ˆ b: ‰O2 Š (Transitive) If a R b and b R c, then a R c: Then R is called a partial order or, simply an order relation, and R is said to de®ne a partial ordering of S The set S with the partial order is called a partially ordered set or, simply, an ordered set or poset We write …S; R† when we want to specify the relation R: The most familiar order relation, called the usual order, is the relation (read ``less than or equal'') on the positive integers N or, more generally, on any subset of the real numbers R For this reason, a partial order relation is usually denoted by P; and aPb is read ``a precedes b'' In this case we also write: a J b means a P b and a Tˆ b; read ``a strictly precedes b'' b Q a means a P b; read ``b succeeds a'' bK a means a J b; read ``b strictly succeeds a'' P =, J =, Q = , and K = are self-explanatory When there is no ambiguity, the symbols Q, respectively , , and ! are frequently used instead of P, J, K, and EXAMPLE 14.1 (a) Let S be any collection of sets The relation  of set inclusion is a partial ordering of S Speci®cally, A  A for any set A; if A  B and B  A then A ˆ B; and if A  B and B  C then A  C (b) Consider the set N of positive integers We say ``a divides b'', written ajb, if there exists an integer c such that ac ˆ b For example, j 4, j 12, j 21, and so on This relation of divisibility is a partial ordering of N (c) The relation ``j'' of divisibility is not an ordering of the set Z of integers Speci®cally, the relation is not antisymmetric For instance, j À and À2 j2, but Tˆ À2 (d ) Consider the set Z of integers De®ne a R b if there is a positive integer r such that b ˆ a r For instance, R since ˆ 23 Then R is a partial ordering of Z Dual Order Let P be any partial ordering of a set S The relation Q, that is, a succeeds b, is also a partial ordering of S; it is called the dual order Observe that a P b if and only if b Q a; hence the dual order Q is the inverse of the relation P, that is, Q ˆ PÀ1 442 Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e CHAP 14] 14 Ordered Sets and Lattices © The McGraw−Hill Companies, 2004 Text ORDERED SETS AND LATTICES 443 Ordered Subsets Let A be a subset of an ordered set S, and suppose a; b P A De®ne a P b as elements of A whenever a P b as elements of S This de®nes a partial ordering of A called the induced order on A The subset A with the induced order is called an ordered subset of S Unless otherwise stated or implied, any subset of an ordered set S will be treated as an ordered subset of S: Quasi-order Suppose J is a relation on a set S satisfying the following two properties: ‰Q1 Š ‰Q2 Š (Irre¯exive) For any a P A, we have a J = a: (Transitive) If a J b, and b J c, then a J c Then J is called a quasi-order on S: There is a close relationship between partial orders and quasi-orders Speci®cally, if P is a partial order on a set S and we de®ne a J b to mean a P b but a Tˆ b, then J is a quasi-order on S Conversely, if J is a quasi-order on a set S and we de®ne a P b to mean a J b or a ˆ b, then P is a partial order on S This allows us to switch back and forth between a partial order and its corresponding quasi-orders using whichever is more convenient Comparability, Linearly Ordered Sets Suppose a and b are elements in a partially ordered set S We say a and b are comparable if aPb or bPa that is, if one of them precedes the other Thus a and b are noncomparable, written akb if neither a P b nor b P a The word ``partial'' is used in de®ning a partially ordered set S since some of the elements of S need not be comparable Suppose, on the other hand, that every pair of elements of S are comparable Then S is said to be totally ordered or linearly ordered, and S is called a chain Although an ordered set S may not be linearly ordered, it is still possible for a subset A of S to be linearly ordered Clearly, every subset of a linearly ordered set S must also be linearly ordered EXAMPLE 14.2 (a) Consider the set N of positive integers ordered by divisibility Then 21 and are comparable since j 21 On the other hand, and are noncomparable since neither j nor j Thus N is not linearly ordered by divisibility Observe that A ˆ f2; 6; 12; 36g is a linearly ordered subset of N since j 6, j 12 and 12 j 36: (b) The set N of positive integers with the usual order ordered subset of N is also linearly ordered (c) The power set P…A† of a set A with two or more elements is not linearly ordered by set inclusion For instance, suppose a and b belong to A Then fag and fbg are noncomparable Observe that the empty set D, fag, and A form a linearly ordered subset of P…A† since D  fag  A: Similarly, D, fbg, and A form a linearly ordered subset of P…A†: (less than or equal) is linearly ordered and hence every Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 14 Ordered Sets and Lattices 444 © The McGraw−Hill Companies, 2004 Text ORDERED SETS AND LATTICES [CHAP 14 Product Sets and Order There are a number of ways to de®ne an order relation on the Cartesian product of given ordered sets Two of these ways follow: (a) Product Order: Suppose S and T are ordered sets Then the following, is an order relation on the product set S  T, called the product order: …a; b†P…a H ; b H † if a a H and b bH (b) Lexicographical Order: Suppose S and T are linearly ordered sets Then the following is an order relation on the product set S  T, called the lexicographical or dictionary order: …a; b† J …a H ; b H † if a