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 Human'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 diculty 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 B if the sets A and B are equal, and we write A T 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 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 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 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 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 fx: x2 À 3x 0g In other words, E consists of those numbers which are solutions of the equation x2 À 3x 0, sometimes called the solution set of the given equation Since the solutions of the equation are and 2, we could also write E f1; 2g (d ) Let E fx: x2 À 3x 0g, F f2; 1g and G f1; 2; 2; 1; 63g Then E F 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 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 fx: x is a positive integer, x2 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 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 f1; 3; 4; 5; 8; 9g B f1; 2; 3; 5; 7g C 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 f2; 4; 6g is a subset of the set F f6; 2; 4g, since each number 2, 4, and belonging to E also belongs to F In fact, E 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 B Conversely, if A 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 B if and only if A B and B A If A B, then it is still possible that A B When A B but A T 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 f1; 3g B f1; 2; 3g; C 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 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 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 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 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 f1; 2; 3; 4g, B f3; 4; 5; 6; 7g, C f2; 3; 5; 7g Then A B f1; 2; 3; 4; 5; 6; 7g A C f1; 2; 3; 4; 5; 7g A B f3; 4g A C 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 MF C since each student in C belongs to either M or F On the other hand, MF D 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 A, and A B 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 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 dierence 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 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 N f1; 2; 3; g, the positive integers, is the universal set Let A f1; 2; 3; 4; g; B f3; 4; 5; 6; 7g; C f6; 7; 8; 9g and let E f2; 4; 6; 8; g, the even integers Then Ac f5; 6; 7; 8; g; B c f1; 2; 8; 9; 10; g; C c f1; 2; 3; 4; 5; 10; 11; g and AnB f1; 2g; c BnC f3; 4; 5g; Also, E f1; 3; 5; g, the odd integers BnA f5; 6; 7g; CnE 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 A B C; P2 A B C c P3 A B c C; P4 A B c C c ; P5 Ac B C; P6 Ac B C c P7 Ac B c C P8 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 Dierence The symmetric dierence 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 A Bn A B One can also show (Problem 1.18) that A È B AnB BnA For example, suppose A f1; 2; 3; 4; 5; 6g and B f4; 5; 6; 7; 8; 9g Then AnB f1; 2; 3g; BnA f7; 8; 9g and so A È B 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 Bc 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) AAA (1b) AAA Associative laws (2a) A B C A B C (3a) ABBA (2b) A B C A B C Commutative laws (3b) ABBA Distributive laws (4a) A B C A B A C (4b) A B C A B A C Identity laws (5a) ADA (6a) AU U (5b) AU A (6b) ADD Involution laws (7) Ac c A Complement laws (8a) A Ac U (8b) A Ac D (9a) Uc D (9b) Dc U DeMorgan's laws (10a) A Bc Ac B c (10b) A Bc Ac B c Method 1: We ®rst show that A Bc Ac B c If x P A Bc , 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 BC 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 Bc We have proven that every element of A Bc belongs to Ac Bc and that every element of Ac Bc belongs to A Bc Together, these inclusions prove that the sets have the same elements, i.e., that A Bc Ac Bc Method 2: From the Venn diagram for A B in Fig 1-4, we see that A Bc 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 Bc 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 A is D A B A 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 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 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 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 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 Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems 374 © The McGraw−Hill Companies, 2004 Text ALGEBRAIC SYSTEMS [CHAP 12 Integers Modulo m Consider the group Z of integers under addition Let H denote the multiples of 5, that is, H fF F F ; À10; À5; 0; 5; 10; F F Fg Then H is a subgroup (necessarily normal) of Z The cosets of H in Z follow: "0 H H fF F F ; À10; À5; 0; 5; 10; F F Fg "0 H fF F F ; À9; À4; 1; 6; 11; F F Fg 2" H fF F F ; À8; À3; 2; 7; 12; F F Fg "3 H fF F F ; À7; À2; 3; 8; 13; F F Fg "4 H fF F F ; À6; À1; 4; 9; 14; F F Fg By the above Theorem 12.8, Z=H f0; 1; 2; 3; 4g is a group under coset addition; its addition table appears in Fig 12-5 This quotient group Z=H is referred to as the integers modulo and is frequently denoted by Z5 Analogously, for any positive integer n, there exists the quotient group Zn called the integers modulo n: Fig 12-5 Cyclic Subgroups Let G be any group and let a be any element of G As usual, we de®ne a0 e and an1 an Á a: Clearly, am Á an amn and am n amn , for any integers m and n All the powers of a, F F F ; aÀ3 ; aÀ2 ; aÀ1 ; e; a; a ; a ; F F F form a subgroup of G called the cyclic group generated by a, and will be denoted by gp a Suppose that the powers of a are not distinct, say ar as with, say, r > s Then arÀs e where r À s > The smallest positive integer m such that am e is called the order of a and will be denoted by jaj: If jaj m, then its cyclic subgroup gp a has m elements given by gp a fe; a; a ; a ; F F F ; a mÀ1 g For example, consider the element 1 in the symmetric group S3 discussed above We have 11 1 ; 21 2 ; 31 2 Á 1 Hence j 1 j and gp 1 fe; 1 ; 2 ; g Observe that j 1 j divides the order of S3 This is true in general; that is, for any element a in a group G we have that jaj equals the order of gp a which divides jGj by Lagrange's Theorem 12.7 We also remark that a group G is said to be cyclic if it has an element a such that G gp a: Generating Sets, Generators Consider any subset A of a group G Let gp A denote the set of all elements x in G such that x is equal to a product of elements where each element comes from the set A AÀ1 (where AÀ1 denotes the set of inverses of elements of A) That is, gp A fx P G: x b1 b2 Á Á Á bm where each bi P A AÀ1 g Then gp A is a subgroup of G with generating set A In particular, A is said to generate the group G if Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems CHAP 12] © The McGraw−Hill Companies, 2004 Text 375 ALGEBRAIC SYSTEMS G gp A, that is, if every g in G is a product of elements from A AÀ1 We say A is a minimal set of generators of G if A generates G and if no set with fewer elements than A generates G For example, the permutations a 1 and b 1 form a minimal set of generators of the symmetric group S3 (Fig 12-4) since: a2 ; 1 a; 2 ab; 3 ab2 ; 1 b; 2 b2 and moreover, S3 is not cyclic and so it cannot be generated by one element Homomorphisms A mapping f from a group G into a group G H is called a homomorphism if f ab f af b for every a; b P G In addition, if f is one-to-one and onto, then f is called an isomorphism and G and G H are said to be isomorphic, written G G H : If f : G G H is a homomorphism, then the kernel of f , written Ker f , is the set of elements whose image is the identity element e H of G H ; that is, Ker f fa P G: f a e H g Recall that the image of f , written f G or Im f , consists of the images of the elements under f ; that is, Im f fb P G H : there exists a P G for which f a bg: The following theorem (proved in Problem 12.21) is fundamental to group theory Theorem 12.9: Let f : G G H be a homomorphism with kernel K: Then K is a normal subgroup of G, and the quotient group G=K is isomorphic to f G: EXAMPLE 12.13 (a) Let G be the group of real numbers under addition, and let G H be the group of positive real numbers under multiplication The mapping f : G G H de®ned by f a 2a is a homomorphism because f a b 2ab 2a 2b f af b In fact, f is also one-to-one and onto; hence G and G H are isomorphic (b) Let G be the group of nonzero complex numbers under multiplication, and let G H be the group of nonzero real numbers under multiplication The mapping f : G G H de®ned by f z jzj is a homomorphism because f z1 z2 jz1 z2 j jz1 j jz2 j f z1 f z2 The kernel K of f consists of those complex numbers z on the unit circle, i.e., for which jzj Thus G=K is isomorphic to the image of f , i.e., to the group of positive real numbers under multiplication (c) Let a be any element in a group G The function f : Z G de®ned by f n an is a homomorphism since f m n amn am Á an f m Á f n The image of f is gp a, the cyclic subgroup generated by a By Theorem 12.9 gp a Z=K where K is the kernel of F If K f0g, then gp a Z On the other hand, if m is the order of a, then K fmultiples of mg, and so gp a Zm In other words, any cyclic group is isomorphic to either the integers Z under addition, or to Zm , the integers under addition modulo m Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems 376 12.6 © The McGraw−Hill Companies, 2004 Text ALGEBRAIC SYSTEMS [CHAP 12 RINGS, INTEGRAL DOMAINS, AND FIELDS Let R be a nonempty set with two binary operations, an operation of addition (denoted by ) and an operation of multiplication (denoted by juxtaposition) Then R is called a ring if the following axioms are satis®ed: R1 R2 R3 R4 R5 R6 For any a; b; c P R, we have a b c a b c: There exists an element P R, called the zero element, such that a a a for every a P R: For each a P R there exists an element Àa P R, called the negative of a, such that a Àa Àa a 0: For any a; b P R, we have a b b a: For any a; b; c P R, we have abc a bc: For any a; b; c P R, we have (i) a b c ab ac, and (ii) b ca ba ca: Observe that the axioms R1 through R4 may be summarized by saying that R is an abelian group under addition Subtraction is de®ned in R by a À b a Àb: It can be shown (see Problem 12.29) that a Á Á a for every a P R A subset S of R is a subring of R if S itself is a ring under the operations in R We note that S is a subring of R if (i) P S, and (ii) for any a; b P S, we have a À b P S and ab P S: Special Kinds of Rings: Integral Domains and Fields This subsection de®nes a number of different kinds of rings, including integral domains and ®elds R is called a commutative ring if ab ba for every a; b P R R is called a ring with an identity element if the element has the property that a Á Á a a for every element a P R In such a case, an element a P R is called a unit if a has a multiplicative inverse, that is, an element aÀ1 in R such that aaÀ1 aÀ1 a 1: R is called a ring with zero divisors if there exist nonzero elements a; b P R such that ab In such a case, a and b are called zero divisors De®nition: A commutative ring R is an integral domain if R has no zero divisors, i.e., if ab implies a or b 0: De®nition: A commutative ring R with an identity element (not equal to 0) is a ®eld if every nonzero a P R is a unit, i.e., has a multiplicative inverse A ®eld is necessarily an integral domain; for if ab and a T 0, then b Á b aÀ1 ab aÀ1 Á We remark that a ®eld may also be viewed as a commutative ring in which the nonzero elements form a group under multiplication EXAMPLE 12.14 (a) The set Z of integers with the usual operations of addition and multiplication is the classical example of an integral domain (with an identity element) The units in Z are only and À1, that is, no other element in Z has a multiplicative inverse (b) The set Zm f0; 1; 2; F F F ; m À 1g under the operation of addition and multiplication modulo m is a ring; it is called the ring of integers modulo m If m is a prime, then Zm is a ®eld On the other hand, if m is not a prime then Zm has zero divisors For instance, in the ring Z6 : 2Á30 but T mod 6 and T (mod 6) Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems CHAP 12] (c) © The McGraw−Hill Companies, 2004 Text 377 ALGEBRAIC SYSTEMS The rational numbers Q and the real numbers R each form a ®eld with respect to the usual operations of addition and multiplication (d ) Let M denote the set of  matrices with integer or real entries Then M is a noncommutative ring with zero divisors under the operations of matrix addition and matrix multiplication M does have an identity element, the identity matrix (e) Let R be any ring Then the set Rx of all polynomials over R is a ring with respect to the usual operations of addition and multiplication of polynomials Moreover, if R is an integral domain then Rx is also an integral domain Ideals A subset J of a ring R is called an ideal in R if the following three properties hold: (i) P J: (ii) For any a; b P J, we have a À b P J: (iii) For any r P R and a P J, we have ra; ar P J: Note ®rst that J is a subring of R Also, J is a subgroup (necessarily normal) of the additive group of R Thus we can form the collection of cosets fa JX a P Rg which forms a partition of R: The importance of ideals comes from the following theorem which is analogous to Theorem 12.7 for normal subgroups Theorem 12.10: Let J be an ideal in a ring R Then the cosets fa J: a P Rg form a ring under the coset operations a J b J a b J and a J b J ab J This ring is denoted by R=J and is called the quotient ring Now let R be a commutative ring with an identity element For any a P R, the set a fra: r P Rg aR is an ideal, it is called the principal ideal generated by a If every ideal in R is a principal ideal, then R is called a principal ideal ring In particular, if R is also an integral domain, then R is called a principal ideal domain (PID) EXAMPLE 12.15 (a) Consider the ring Z of integers Then every ideal J in Z is a principal ideal, that is, J m mZ, for some integer m Thus Z is a principal ideal domain (PID) The quotient ring Zm Z= m is simply the ring of integers modulo m Although, Z is an integral domain (no zero divisors), the quotient ring Zm may have zero divisors, e.g., and are zero divisors in Z6 (b) Let R be any ring Then f0g and R are ideals In particular, if R is a ®eld, then f0g and R are the only ideals (c) Let K be a ®eld Then the ring Kx of polynomials over K is a PID (principal ideal domain) On the other hand, the ring Kx; y of polynomials in two variables is not a PID (d ) Let M be the ring of  matrices with integer entries Let J consists of all matrices of the form a b Note that: (i) P J (ii) For any a; b P J, we have a À b P J (iii) For any r P M and a P J, we have P J; that is, RJ J However, JR T J Thus J is not an ideal (It is called a left ideal.) Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems 378 © The McGraw−Hill Companies, 2004 Text ALGEBRAIC SYSTEMS [CHAP 12 Ring Homomorphisms A mapping f from a ring R into a ring R H is called a ring homomorphism or, simply, homomorphism if f a b f a f b, f ab f af b for every a; b P R In addition, if f is one-to-one and onto, then f is called an isomorphism and R and R H are said to be isomorphic, written R R H : If f : R R H is a homomorphism, then the kernel of f , written Ker f , is the set of elements whose image is the zero element of R H ; that is, Ker f fr P R: f r 0g The following theorem (analogous to Theorem 12.9 for groups) is fundamental to ring theory Let f : R R H be a ring homomorphism with kernel K Then K is an ideal in R, and the quotient ring R=K is isomorphic to f R: Theorem 12.11: Divisibility in Integral Domains Now let D be an integral domain We say that b divides a in D if a bc for some c P D An element u P D is called a unit if u divides 1, i.e., if u has a multiplicative inverse An element b P D is called an associate of a P D if b ua for some unit u P D A nonunit p P D is said to be irreducible if p ab implies a or b is a unit An integral domain D is called a unique factorization domain (UFD) if every nonunit a P D can be written uniquely (up to associates and order) as a product of irreducible elements EXAMPLE 12.16 (a) (b) The ring Z of integers is the classical example of a unique factorization domain The units of Z are and À1 The only associates of n P Z are n and Àn The irreducible elements of Z are the prime numbers p p The set pD fa b 13 X a; b integersg p is an integral p domain The units of D are Ỉ1, 18 Ỉ 13 and À18 Ỉ 13 The elements 2, À 13 and À3 À 13 are irreducible in D Observe that p p Á 3 À 13 À3 À 13 Thus D is not a unique factorization domain (See Problem 12.99.) 12.7 POLYNOMIALS OVER A FIELD This section investigates polynomials whose coef®cients come from some integral domain or ®eld K In particular, we show that polynomials over a ®eld K have many of the same properties as the integers Basic De®nitions Let K be an integral domain or a ®eld Formally, a polynomial f over K is an in®nite sequence of elements from K in which all except a ®nite number of them are 0; that is, f F F F ; 0; an ; F F F ; a1 ; a0 or, equivalently, f t an t n Á Á Á a0 where the symbol t is used as an indeterminate The entry ak is called the kth coecient of f If n is the largest integer for which an T 0, then we say that the degree of f is n, written deg f n We also call an the leading coecient of f , and, if an 1, we call f a monic polynomial On the other hand, if every coecient of f is then f is called the zero polynomial, written f The degree of the zero polynomial is not de®ned Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems CHAP 12] © The McGraw−Hill Companies, 2004 Text 379 ALGEBRAIC SYSTEMS Let Kt be the collection of all polynomials f t over K Addition and multiplication are de®ned in Kt as follows Suppose f t an t n Á Á Á a0 g t bm t m Á Á Á b0 and Then the sum f g is the polynomial obtained by adding corresponding coecients; that is, if m n, then f t g t an t n Á Á Á am bm t m Á Á Á a1 b1 t a0 b0 Furthermore, the product of f and g is the polynomial f tg t an bm t nm Á Á Á a1 b0 a0 b1 t a0 b0 That is, f tg t cnm t nm Á Á Á c1 t c0 where ck k X i0 bkÀ1 a0 bk a1 bkÀ1 Á Á Á ak b0 The set K of scalars is viewed as a subset of Kt Speci®cally, we identify the scalar a0 P K with the polynomial f t a0 or a0 F F F ; 0; 0; a0 Then the operators of addition and scalar multiplication are preserved by this identi®cation; that is, F F F ; 0; a0 F F F ; 0; b0 F F F ; 0; a0 b0 and F F F ; 0; a0 Á F F F ; 0; a0 b0 F F F ; 0; a0 b0 Thus, the mapping É: K Kt de®ned by É a0 a0 is an isomorphism which embeds K into Kt: Theorem 12.12: Let K be an integral domain The Kt under the operations of addition and multiplication of polynomials is a commutative ring with an identity element The following simple result has important consequences Lemma 12.13: Suppose f and g are polynomials over an integral domain K Then deg fg deg f deg g The proof follows directly from the de®nition of the product of polynomials That is, suppose f t an t n Á Á Á a0 and g t bm t m Á Á Á b0 , where an T and bm T 0: Then f tg t an bm t mn terms of lower degree Also, since K is an integral domain with no zero divisors, an bm T Thus and the lemma is proved deg fg m n deg f deg g The following proposition lists many properties of our polynomials [Recall that a polynomial g is said to divide a polynomial f if there exists a polynomial h such that f t g th t: Proposition 12.14: Let K be an integral domain and let f and g be polynomials over K (i) Kt is an integral domain (ii) The units of Kt are the units in K: (iii) If g divides f , then deg g deg f or f 0: (iv) If g divides f and f divides g, then f t kg t where k is a unit in K: (v) If d and d H are monic polynomials such that d divides d H and d H divides d, then d d H: Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems 380 © The McGraw−Hill Companies, 2004 Text ALGEBRAIC SYSTEMS [CHAP 12 Euclidean Algorithm, Roots of Polynomials This subsection discusses the roots of a polynomial f t, where we now assume the coef®cients of f t come from a ®eld K Recall that a scalar a P K is a root of a polynomial f t if f a First we begin with an important theorem which is very similar to a corresponding theorem for the integers Z Theorem 12.15 (Euclidean Division Algorithm): Let f t and g t be polynomials over a ®eld K with g t T Then there exist polynomials q t and r t such that f t q tg t r t where either r t or deg r < deg g: The above theorem (Proved in Problem 12.39) formalizes the process known as ``long division'' The polynomial q t is called the quotient and the polynomial r t is called the remainder when f t is divided by g t Corollary 12.16 (Remainder Theorem): remainder Suppose f t is divided by g t t À a Then f a is the The proof follows from the Euclidean Algorithm That is, dividing f t by t À a we get f t q t t À a r t where deg r < deg t À a Hence r t r is a scalar Substituting t a in the equation for f t yields f a q a a À a r q t Á r r Thus f a is the remainder, as claimed Corollary 12.16 also tells us that f a if and only if the remainder r r t Accordingly: Corollary 12.17 (Factor Theorem): f t: The scalar a P K is a root of f t if and only if t À a is a factor of The next theorem tells us the number of possible roots of a polynomial Theorem 12.18: Suppose f t is a polynomial over a ®eld K, and deg f n Then f t has at most n roots The following theorem is the main tool for ®nding rational roots of a polynomial with integer coef®cients Theorem 12.19: Suppose a rational number p=q (reduced to lowest terms) is a root of the polynomial f t an t n Á Á Á a1 t a0 where all the coecients an ; F F F ; a1 ; a0 are integers Then p divides the constant term a0 and q divides the leading coecient an : In particular, if c p=q is an integer, then c divides the constant term a0 : EXAMPLE 12.17 (a) Suppose f t t3 t2 À 8t 4: Assuming f t has a rational root, ®nd all the roots of f t Since the leading coecient is 1, the rational roots of f t must be integers from among Æ1, Æ2, Æ4 Note f 1 T and f À1 T By synthetic division, or dividing by t À 2, we get À À 4 À Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems CHAP 12] © The McGraw−Hill Companies, 2004 Text 381 ALGEBRAIC SYSTEMS Therefore t is a root and f t t À 2 t2 3t À 2 Using the quadratic formula for t2 3t À 0, we obtain the following roots of f t: p p t 2; t À3 17=2; t À3 À 17=2 (b) Suppose h t t4 À 2t3 11t À 10 Find all the real roots of h t assuming there are two integer roots The integer roots must be among Ỉ1, Ỉ2, Ỉ5, Ỉ10 By synthetic division (or dividing by t À and then t 2) we get 1 À À À 11 À 10 10 À2 À À À À 10 10 À Thus t and t À2 are roots and h t t À 1 t 2 t À 3t 5 The quadratic formula with t2 À 3t tells us that there are no other real roots That is, t and t À2 are the only real roots of h t: K[t] as a PID and UFD The following theorems apply Theorem 12.20: The ring Kt of polynomials over a ®eld K is a principal ideal domain (PID) If J is an ideal in Kt, then there exists a unique monic polynomial d which generates J, that is, every polynomial f in J is a multiple of d Theorem 12.21: Let f and g be polynomials in Kt, not both zero Then there exists a unique monic polynomial d such that: (i) d divides both f and g (ii) If d H divides f and g, then d H divides d The polynomial d in the above theorem is called the greatest common divisor of f and g, written d gcd f ; g If d 1, then f and g are said to be relatively prime Corollary 12.22: Let d be the greatest common divisor of f and g Then there exist polynomials m and n such that d mf ng In particular, if f and g are relatively prime, then there exist polynomials m and n such that mf ng 1: A polynomial p P Kt is said to be irreducible if p is not a scalar and if p fg implies f or g is a scalar In other words, p is irreducible if its only divisors are its associates (scalar multiples) Lemma 12.23: Suppose p P Kt is irreducible If p divides the product fg of polynomials f and g in Kt, then p divides f or p divides g More generally, if p divides the product f1 f2 Á Á Á fn of n polynomials, then p divides one of them The next theorem states that the polynomials over a ®eld form a unique factorization domain (UFD) Theorem 12.24 (Unique Factorization Theorem): Let f be a nonzero polynomial in Kt Then f can be written uniquely (except for order) as a product f kp1 p2 Á Á Á pn where k P K and the p H s are monic irreducible polynomials in Kt: Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems 382 © The McGraw−Hill Companies, 2004 Text ALGEBRAIC SYSTEMS [CHAP 12 Fundamental Theorem of Algebra The proof of the following theorem lies beyond the scope of this text Fundamental Theorem of Algebra: Any nonzero polynomial f t over the complex ®eld C has a root in C Thus f t can be written uniquely (except for order) as a product f t k t À r1 t À r2 Á Á Á t À rn where k and the ri are complex numbers and deg f n: The above theorem is certainly not true for the real ®eld R For example, f t t2 is a polynomial over R, but f t has no real root The following theorems apply Theorem 12.25: Suppose f t is a polynomial over the real ®eld R, and suppose the complex number z a bi, b T 0, is a root of f t Then the complex conjugate z" a À bi is also a root of f t Hence c t t À z t À z" t2 À 2at a2 b2 is a factor of f t: Theorem 12.26: Let f t be a nonzero polynomial over the real ®eld R Then f t can be written uniquely (except for order) as a product f t kp1 tp2 t Á Á Á pm t where k P R and the pi t are real monic polynomials of degree or EXAMPLE 12.18 Let f t t4 À 3t3 6t2 25t À 39 Find all the roots of f t given that t 3i is a root Since 3i is a root, then À 3i is a root and c t t2 À 4t 13 is a factor of f t Dividing f t by c t we get f t t2 À 4t 13 t2 t À 3 The quadratic formula with t2 t À gives us the other roots of f t That is, the four roots of f t are as follows: p p t 3i; t À 3i; t À1 13=2; t À1 À 13=2 Solved Problems OPERATIONS AND SEMIGROUPS 12.1 Consider the set N of positive integers, and let à denote the operation of least common multiple lcm on N (a) Find à 6, à 5, à 18, and à (b) Is N; à a semigroup? Is it commutative? (c) Find the identity element of Ã: (d ) Which elements in N, if any, have inverses and what are they? (a) Since x à y means the least common multiple of x and y, we have: à 12; (b) à 15; à 18 18; 1Ã66 One proves in number theory that a à b à c a à b à c, i.e., that the operation of lcm is associative, and that a à b b à a, i.e., that the operation of lcm is commutative Hence N; à is a commutative semigroup Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems CHAP 12] (c) © The McGraw−Hill Companies, 2004 Text 383 ALGEBRAIC SYSTEMS The integer is the identity element since lcm 1; a a for any positive integer a, i.e., à a a à a for any a P N: (d ) Since lcm a; b if and only if a and b 1, the only number which has an inverse is and it is its own inverse 12.2 Consider the set Q of rational numbers, and let à be the operation on Q de®ned by a à b a b À ab (a) Find à 4; à À5 and à 2: (b) Is Q; à a semigroup? Is it commutative? (c) Find the identity element for à (d ) Do any of the elements in Q have an inverse? What is it? (a) à À Á 4 À 12 À5: à À5 À5 À Á À5 À 10 7: à 12 12 À 7 12 4: (b) We have: a à b à c a b À ab à c a b À ab c À a b À abc a b À ab c À ac À bc abc a b c À ab À ac À bc abc a à b à c a à b c À bc a b c À bc À a b c À bc a b c À bc À ab À ac abc Hence à is associative and Q; à is a semigroup Also, a à b a b À ab b a À ba b à a Hence Q; à is a commutative semigroup (c) An element e is an identity element if a à e a for every a P Q Compute as follows: a à e a; a e À ae a; e À ea 0; e 1 À a 0; e0 Accordingly, is the identity element (d ) In order for e to have an inverse x, we must have a à x since is the identity element by Part e Compute as follows: a à x 0; a x À ax 0; a ax À x; a x a À 1; x a= a À 1 Thus if a T 1, then a has an inverse and it is a= a À 1: 12.3 Let S be a semigroup with identity e, and let b and b H be inverses of a Show that b b H , that is, that inverses are unique if they exist We have: b à a à b H b à e b and b à a à b H e à b H b H Since S is associative, b à a à b H b à a à b H ; hence b b H : Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems 384 12.4 © The McGraw−Hill Companies, 2004 Text ALGEBRAIC SYSTEMS [CHAP 12 State whether or not each of the following six subsets of the positive integers N is closed under the operation of multiplication: (a) A f0; 1g: (d ) D f2; 4; 6; F F Fg fx: x is eveng (b) B f1; 2g: (e) (c) C fx: x is primeg ( f ) F f2; 4; 8; F F Fg x: x 2n ; n P Ng: E f1; 3; 5; F F Fg fx: x is oddg: Which of the six sets, if any, are closed under the operation of addition? (a) We have: Á 0; Á 0; Á 0; 1Á11 Hence A is closed under multiplication (b) Since Á 4, which does not belong to B, the set B is not closed under multiplication (c) Note that and are prime but Á is not prime; hence C is not closed under multiplication (d ) The product of even numbers is even; hence D is closed under multiplication (e) The product of odd numbers is odd; hence E is closed under multiplication ( f ) Since 2r Á 2s 2rs , F is closed under multiplication Since the sum of two even integers is even, the set D is closed under addition However, each of the other sets is not closed under addition since, for example, 12.5 112P = A; 358P = C; 123P = B; 134P =E 246P =F Let S N  N Let à be the operation on S de®ned by a; b à a H ; b H aa H ; bb H : (a) Show that à is associative (Hence S is a semigroup.) (b) De®ne f : S; à Q;  by f a; b a=b Show that f is a homomorphism (c) Find the congruence relation $ in S determined by the homomorphism f , i.e., x $ y if f x f y (See Theorem 12.4.) (d ) Describe S= $ Does S= $ have an identity element? Does it have inverses? Suppose x a; b; y c; d; z e; f : (a) We have xyz ac; bd à e; f ace; bdf x yz a; b à ce; df a ce; b df Since a; b; c; d; e; f , are positive integers, ace a ce and bdf b df Thus xyz x yz and hence à is associative That is, S; à is a semigroup (b) We have: f x à y f ac; bd ac= bd a=b c=d f xf y Thus f is a homomorphism (c) Suppose f x f y Then a c and hence ad bc: b d Thus f determines the congruence relation $ on S de®ned by a; b $ c; d if ad bc (d ) The image of f is Q , the set of positive rational numbers By Theorem 12.3, S= $ is isomorphic to Q Thus S= $ does have an identity element, and every element has an inverse Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems CHAP 12] 12.6 © The McGraw−Hill Companies, 2004 Text 385 ALGEBRAIC SYSTEMS Let S N  N Let à be the operation on S de®ned by a; b à a H ; b H a a H ; b b H (a) Show that à is associative (Hence S is a semigroup.) (b) De®ne f : S; à Z; by f (a,b)=a À b Show that f is a homomorphism (c) Find the congruence relation $ in S determined by the homomorphism f , i.e., x $ y if f x f y (See Theorem 12.4.) (d ) Describe S= $ Does S= $ have an identity element? Inverses? Suppose x a; b; y c; d, z e; f : (a) We have xyz a c; b d à e; f a c e; b d f x yz a; b à c e; d f a c e; b d f Since a; b; c; d; e; f are positive integers, a c e a c e and b d f b d f Thus xyz x yz and hence à is associative That is, S; à is a semigroup (b) We have: f x à y f a c; b d a c À b d a À b c À d f xf y Thus f is a homomorphism (c) Suppose f x f y Then a À b c À d and hence a d b d Thus f determines the congruence relation $ on S de®ned by a; b $ c; d if a d b c (d ) The image of f is all of Z since every integer is the dierence of two positive integers Therefore, by Theorem 12.3, S= $ is isomorphic to Z Thus S= $ does have an identity element, and every element has an (additive) inverse 12.7 Prove Theorem 12.1: Suppose à is an associative operation on a set S Then any product a1 à a2 à Á Á Á à an requires no parenthesis, that is, all possible products are equal The proof is by induction on n Since n is associative, the theorem holds for n 1; 2, and Suppose n ! We use the notation: a1 a2 ; Á Á Á an Á Á Á a1 a2 a3 Á Á Áan and a1 a2 Á Á Á an any product We show a1 a2 Á Á Á an a1 a2 Á Á Á an and so all such products will be equal Since a1 a2 Á Á Á an denotes some product, there exists an r < n such that a1 a2 Á Á Á an a1 a2 Á Á Á ar ar1 Á Á Á an Therefore, by induction, a1 a2 Á Á Á an a1 a2 Á Á Á ar ar1 Á Á Á an a1 a2 Á Á Á ar ar1 Á Á Á an a1 Á Á Á ar ar1 Á Á Á anÀ1 an a1 Á Á Á ar arÀ1 Á Á Á anÀ1 an a1 Á Á Á anÀ1 an a1 Á Á Á anÀ1 an a1 a2 Á Á Á an Thus the theorem is proved 12.8 Prove Theorem 12.4: Let f : S S H be a semigroup homomorphism Let a $ b if f a f b Then: (i) $ is a congruence relation, (ii) S= $ is isomorphic to f S: (i) First we show that $ is an equivalence relation Since f a f a, we have a $ a If a $ b, then f a f b or f b f a and hence b $ a Lastly, if a $ b and b $ c, then f a f b and f b f c, and hence f a f c Thus a $ c That is, $ is an equivalence relation Suppose now a $ a H and b $ b H Then f a f a H and f b f b H Since f is a homomorphism, Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems 386 © The McGraw−Hill Companies, 2004 Text ALGEBRAIC SYSTEMS [CHAP 12 f ab f a f b f a H f b H f a H b H Therefore ab $ a H b H That is, $ is a congruence relation (ii) De®ne É: S= $3 f S by É a f a We need to prove: (1) É is well-de®ned, that is, É a P f S, and if a b then f a f b (2) É is an isomorphism, that is, É is a homomorphism, one-to-one and onto (1) Proof that É is well-de®ned: We have É a f a Since a P S, we have f a P f S Hence É a P f S, as required Now suppose a b Then a $ b and hence f a f b Thus É a f a f b É b That is, É is well-de®ned (2) Proof that É is an isomorphism: Since f is a homomorphism, É ab Éab f ab f af b É aÉ b Hence É is a homomorphism Suppose É a É b Then f a f b, and so a $ b Thus a b and É is one-to-one Lastly, let y P f S Then, f a y for some a P S Hence É a f a y Thus É is onto f S Accordingly, É is an isomorphism GROUPS 12.9 Consider the symmetric group S3 whose multiplication table is given in Fig 12-4 (a) Find the order and the group generated by each element of S3 : (b) Find the number and all subgroups of S3 : (c) Let A f1 ; 2 g and B f 1 ; 2 g Find AB, 3 A, and A3 (d ) Let H gp 1 and K gp 2 Show that HK is not a subgroup of S3 : (e) Is S3 cyclic? (a) There are six elements: (1) , (2) 1 , (3) 2 , (4) 3 ; (5) 1 , (6) 2 Find the powers of each element x until xn Then jxj n, and gp x f; x; x2 ; F F F ; xnÀ1 g: Note x1 x, so we need only begin with n when x T : (1) 1 ; so jj and g fg: (2) 21 ; hence j1 j and gp 1 f; 1 g (3) (4) (5) (6) (b) 22 ; hence j2 j and gp 2 f; 2 g: 23 ; hence j3 j and gp 3 f; 3 g: 21 2 ; 31 2 1 ; hence j 1 j and gp 1 f; 1 ; 2 g: 22 1 , 32 1 2 ; hence j 2 j and gp 1 f; 2 ; 1 g: First of all, H1 feg and H2 S3 are subgroups of S3 Any other subgroup of S3 must have order or since its order must divide S3 Since and are prime numbers, these subgroups must be cyclic (Problem 12.65) and hence must appear in part (a) Thus the other subgroups of S3 are H3 fe; 1 g; H4 fe; 2 g, H5 fe; 3 g, H6 fe; 1 ; 2 g Accordingly, S3 has six subgroups (c) Multiply each element of A by each element of B: 1 2 ; 1 3 , 3 , 1 3 2 1 Hence AB f1 ; 2 ; 3 g: Multiply 3 by each element of A: 3 1 , 3 2 , hence 3 A f 1 ; 2 g Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems CHAP 12] © The McGraw−Hill Companies, 2004 Text 387 ALGEBRAIC SYSTEMS Multiply each element of A by 3 : 1 3 , 2 3 , hence A3 f 1 ; 2 g (d ) H fe; 1 g; K fe; 2 g and then HK fe; 1 ; 2 ; 1 g, which is not a subgroup of S3 since HK has four elements (e) S3 is not cyclic since S3 is not generated by any of its elements 12.10 Consider the group G f1; 2; 3; 4; 5; 6g under multiplication modulo (a) Find the multiplication table of G (b) Find 2À1 ; 3À1 ; 6À1 : (c) Find the orders and subgroups generated by and (d ) Is G cyclic? (a) To ®nd a à b in G, ®nd the remainder when the product ab is divided by For example, Á 30 which yields a remainder of when divided by 7; hence à in G The multiplication table of G appears in Fig 12-6 (b) Note ®rst that is the identity element of G Recall that aÀ1 is that element of G such that aaÀ1 Hence 2À1 4, 3À1 and 6À1 6: (c) We have 21 2, 22 4, but 23 Hence j2j and gp 2 f1; 2; 4g We have 31 3, 32 2, 33 6, 34 4, 35 5, 36 Hence j3j and gp 3 G: (d ) G is cyclic since G gp 3: Fig 12-6 12.11 Let G be a reduced residue system modulo 15, say, G f1; 2; 4; 7; 8; 11; 13; 14g (the set of integers between and 15 which are coprime to 15) Then G is a group under multiplication modulo 15 (a) Find the multiplication table of G (b) Find 2À1 , 7À1 , 11À1 : (c) Find the orders and subgroups generated by 2, 7, and 11 (d ) Is G cyclic? (a) To ®nd a à b in G, ®nd the remainder when the product ab is divided by 15 The multiplication table appears in Fig 12-7 (b) The integers r and s are inverses if r à s Hence 2À1 8, 7À1 13, 11À1 11 Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 12 Algebraic Systems 388 © The McGraw−Hill Companies, 2004 Text ALGEBRAIC SYSTEMS (c) [CHAP 12 We have 22 4, 23 8, 24 Hence j2j and gp 2 f1; 2; 4; 8g Also, 72 4, 73 à 13, 74 13 à Hence j7j and gp 7 f1; 4; 7; 13g: Lastly, 112 Hence j11j and gp 11 f1; 11g: (d ) No, since no element generates G Fig 12-7 12.12 Let and be the following elements of the symmetric group S6 : 3 5 6 and 3 6 Find: , , 2 , and À1 : (Since and are functions, means apply and then .) The eect of and then on 1; 2; F F F ; is pictured in Fig 12-8(a); the eect of and then on 1; 2; F F F ; is pictured in Fig 12-8(b); and the eect of and then again on 1; 2; F F F ; is pictured in Fig 12-8(c) Thus 6 ; 2 6 We obtain À1 by interchanging the top and bottom rows of and then rearranging: 2 À1 6 Fig 12-8 12.13 Let S be the square in the plane R2 pictured in Fig 12-9, with its center at the origin Note that the vertices of S are numbered counterclockwise from to (a) De®ne the group G of symmetries of S (b) List the elements of G (c) Find a minimum set of generators of G: (a) A symmetry of S is a rigid one-to-one correspondence between S and itself (Here rigid means that distances between points not change.) The group G of symmetries of S is the set of all symmetries of S under composition of mappings