should be 9 12 1 Terms Theorems and Algorithms 581. p xviii Appendix A 587,should be Appendix A Languages and Regular Sets. p xviii Appendix B 591,should be Appendix B Finite Automata. p xviii Add below Appendix B entry B 1 Exercises,p xx l 20 with the all the. should be with all the,p 5 l 27 Such sets are called a finite set. should be Such a set is called a finite set, p 7 l 8 The bold on A B if and only if A B and B A should be removed. p 10 l 22 We need to show that there is some fixed integer of the form 2k0 3. for k0 N that can be written as j 2 3 for some choice of j N If such a. j existed we would have 2k0 3 j 2 3 In the case k0 0 the element j. would have to satisfy 3 j 2 3 or j 2 6 0 for 3 to be in B. Is better written as We must prove 3 6 B by showing 3 cannot be. written as j 2 3 for any j N If 3 B there exists a j0 N such that. 3 j02 3 If such a j0 existed we would have 3 j02 3 or j02 6 Since. no such j0 exists 3 6 B, p 10 l 27 One set can be a member of a second subset but not every element of. the first set need be an element of the second set. should be One set can be a member of a second subset but not every. element of the second set need be an element of the first set. p 16 l 29 Add line starting at left margin before end of theorem mark d This. part is left as an exercise for the reader, p 18 l 25 Place A under the first figure and B under the second. p 19 l 19 of both B and C,should be of either B or C. p 21 l 4 sometimes call the,should be sometimes called the. p 22 l 2 But then,should be Then, p 23 l 17 There should be a blank line after the end of theorem mark. p 24 l 16 line should be aligned with the left margin and not indented. p 25 l 11 its converse its inverse and its contrapositive. should be converse if b then a its inverse if not a then not b and. its contrapositive if not b then not a,p 25 l 32 Equivalent statements. should be equivalent statements a is true if and only if b is true. p 27 l 17 Theorem 7 b,should be Theorem 7 b in Section 1 3 2. p 28 l 23 When n copies of the same set are used the resulting Cartesian product. X X X is the set of all ordered n tuples of elements in X denoted X n. should be When n copies of the same set are used the product is called. a Cartesian product The resulting Cartesian product X X X is the. set of all ordered n tuples of elements in X denoted X n. p 28 l 27 1 b,should be 1 b, p 28 l 27 Omit the sentence The product of two sets is sometimes referred to. as the Cartesian product, p 28 l 33 l 34 remove italics from the two italicized copies of combinatorial. p 29 l 18 The Absorption Law holds,should be The Absorption Law for Join holds. p 29 l 19 Add The Absorption Law for Meet follows from Theorems 2 and 3 of. Section 1 3 1,p 28 l 32 design that is of a computer. should be design of a computer chip, p 32 l 27 In parts b m sentence should align at the left margin. p 35 l 4 5 6 The markers i ii and iii should be enclosed in parentheses. p 35 l 17 part c,should be Theorem 1 c,p 35 l 18 part a. should be Theorem 1 a,p 35 l 19 part b,should be Theorem 1 b. p 35 l 25 part a,should be Theorem 1 a,p 35 l 27 part b. should be Theorem 1 b,p 37 l 10 Theorem 6,should be Theorem 6 a. p 40 l 3 add Theorem 3 after Three Sets,p 46 l 4 four should be replaced with 4. p 46 l 7 three should be replaced with 3,p 46 l 10 two should be replaced with 2. p 46 l 13 one should be replaced with 1,p 46 l 16 zero should be replaced with 0. p 47 l 20 Let n be any natural number for which the result is true. should be Assume the result is true for some natural number say n. p 49 l 15 Let,should be Assume, p 53 l 18 remove sentence The next result was referred to in the discussion of. computer switches in Section 1 3 4 and will be proved several times in the book. using several different ideas, replace with If you think of the elements of a set as switches that each. can be set on or off the next result tells you how many different ways you can. choose a subset of switches such that each element is on. p 53 l 21 Remove The proof of,p 57 l 8 to be either. should be as either,p 57 l 40 as algorithm,should be as an algorithm. p 58 l 29 algorithm,should be algorithm,p 60 l 27 algorithm. should be algorithm, p 68 bottom part of Figure 1 18 should have no n0 1 n0 2 n0 3 and n0 4. in the five boxes inside the large bottom box,p 68 l 9 p1 p2 pk. should be p1 p2 pk,p 68 l 15 16 17 there are five occurrences of m. should be replace occurrences of m with l, n k l where k 6 1 and l 6 1 It follows easily that 1 k n and that. 1 l n Hence by the inductive hypothesis k l T So k and l can be. p 72 l 1 5 7 remove 8 from the three lists,p 72 l 17 l 8. should have added l 8 where k represents the number of 3 cent stamps. and l the number of 8 cent stamps used to make n cents of postage. p 74 l 12 This case,should be The proof,p 76 l 21 remove do from the end of the line. p 93 l 5 Base cases,should be Base cases,p 93 l 6 Closure rules. should be Closure rules,p 93 l 518 Base cases,should be Base cases. p 93 l 19 Closure rules,should be Closure rules,p 93 l 29 remove after Theorem 1 8. p 95 l 18 p q r p,should be p q r p,p 98 l 10 for these gates. should be for and gates,p 98 l 12 AN D OR,should be OR AN D. p 98 l 22 in Figure 2 7 there are four occurrences of P that should be occurrences. p 99 l 1 In Figure 2 8 the labels A and B should be positioned over the symbols. for the gates as C is and not labelling inputs,p 103 l 19 p q and r. should be p q r,p 105 l 26 p qr p, should be p q r p and the truth values should be centered under. the rightarrow in r p,p 108 l 32 q and r,should be q and r. p 110 l 15 This Proof is left for Exercise 23,should be This is Exercise 23. p 111 l 17 if and only if,should be then,p 111 l 21 Theorem 2 5. should be Theorem 3,p 113 l 13 Half adder,should be Two binary bits are added. p 115 l 14 be easier than another for someone reading the program to under. should be easier to understand for someone reading the program. p 116 l 13 1 should be 1 l 16 2 should be 2 l 21 3 should be 3. p 119 l 11 in the text,should be in the text Figure 2 11. p 120 l 8 the symbol should be the symbol, p 120 ll 28 40 This material should be placed on p 190 as it deals with equiva. lence relations that have not yet been introduced,p 121 l 30 from the other form. should be from any other form,p 123 l 34 Table 2 8 True Terms l2 p q r. should be Table 2 8 True Terms l4 p q r,p 123 l 35 in the interpretations l3 p q r. should be in the interpretations l5 p q r, p 125 figure after l 3 the output of the top negation should not be x but x and. the output from the bottom negation should not be y but y. p 131 l 2 Example 12 above,should be Example 12,p 135 l 11 A formula such as x 3. should be A mathematical formula such as x 3,p 135 l 31 such as. should be such as the mathematical symbol for less than. p 135 l 32 from the universal set,should be from a universal set defined on R. p 136 l 33 Add as a new paragraph just before l 33 that starts with Let i j N. When we allocate storage for an array in a programming language we have. an example of restricted quantification where the universal set is the set of all. memory locations accessible by a program We want to explore this idea about. allocating memory in the next example,p 138 l 4 Example 4 in Section 2 7 4 shows. should be Example 4 shows,p 138 l 19,x y P x y is logically equivalent to x y P x y. x y P x y is logically equivalent to x y P x y,p 138 l 31. d x y x y z x z z y,d x y x y z x z z y,p 139 l 12. x y x y x x z x y,x y x y z x z x y,p 140 l 8 Because should be Since. p 140 l 10 Remove sentence How about x 2 2 2 3 1,p 140 l 29 that is that x and y. should be that is x and y,x P x Q x x P x x Q x,x P x Q x x P x x Q x. p 141 l 14 one cannot check,should be one cannot use testing to check. p 141 l 30 for limit N 2 down to 0,should be for limit N 2 down to 0 do. p 141 l 31 for position 0 up to limit,should be for position 0 up to limit do. p 142 l 24 for limit N 2 down to 0,should be for limit N 2 down to 0 do. p 142 l 28 for position 0 up to limit,should be for position 0 up to limit do. p 145 l 44 d x y P x y Q x y zR x z,should be d x y P x y Q x y z R x z. p 146 l 23 for i 1 2 n,should be for i 0 to n 1 do. p 147 l 4 For t 1 to 2N 1,should be For t 1 to 2N 1 do. p 147 l 6 For position 1 to 2N 1,should be For position 1 to 2N 1 do. p 148 l 13 2 9 1 Terms and Theorems,should be Terms Theorems and Algorithms. p 151 l 24 TRUE and FALSE,should be T RU E and F ALSE. p 152 l 12 TRUE and FALSE,should be T RU E and F ALSE. p 159 l 1 A binary relation is a set of ordered pairs. should be Let X be a set,p 159 l 1 on a set X,should be on X. p 162 l 25 X1 X2 Xn,should be X1 X2 Xn,p 163 l 7 such that for any n1 n2 n3 A. should be for any n1 n2 n3 A, p 173 l 28 How is the relation IsAdjacentT o related to the relation InF rontOf. A person x is adjacent to a person y if x is the person in front of y or y is the. person after x Said another way x IsAdjacentT o y means that x is just in. front of or just behind y, should be A person x is adjacent to a person y if x is the person in front. of y or y is the person after x Said another way x IsAdjacentT o y means that. x is just in front of or just behind y How is the relation IsAdjacentT o related. to the relation InF rontOf,p 175 l 6 By Theorem 1 a relation. should be By Theorem 1 Section 3 4 1 a relation,p 175 l 21 reflexive and transitive closure. should be reflexive and transitive closure, p 185 ll 29 31 from which one may read off the equivalence relation Theorem. 3 says that one can go from a partition to an equivalence relation from. should be from which one may read off the equivalence relation Theorem. 3 says that one can go from a partition to an equivalence relation from. p 186 l 23 By Theorem 3 in this section this relation. should be By Theorem 3 this relation,p 186 l 33 Figure 3 9 on page 187 is a Venn. should be Figure 3 9 is a Venn,p 189 l 30 Prove Theorem 1. should be Prove Theorem 1 in Section 3 6 1,p 190 l 1 Prove Theorem 4. should be Prove Theorem 4 in Section 3 6 2, p 191 l 12 reflexive transitive antisymmetric relation. should be reflexive transitive and antisymmetric relation. p 194 l 2 U V X,should be U V X,p 194 l 5 U V W X,should be U V W X. p 194 ll 19 21 linear ordering or total ordering on X. should be linear ordering or total ordering on X,p 196 ll 4 5 set inclusion P X then 0 1. should be set inclusion on P X then 0 1,p 198 l 7 finite set X and let x X. should be finite set X and let x X, p 200 l 23 Partial order would be better as Partial order schedule for file. processing and email checking Possibly should be Figure designation and call. p 201 l 1 Linear order would be better as Partial order schedule embedded. in linear order,p 201 l 18 Examples 5 a and b are partial. should be Examples 5 a and b in Section 3 8 1 are partial. p 202 l 12 Prove Theorem 1 a,should be Prove Theorem 1 a in Section 3 8 3. p 202 l 1226 Prove Theorem 2,should be Prove Theorem 2 in Section 3 8 4. p 206 l 13 in the relation R0 in Table 3 14,should be in the relation R0 in Table 3 15. p 209 First table labelled R S should be labelled R0. p 209 l 6 of text not in a box join of R and S on N ame. should be natural join of R and S on N ame, p 209 Second table labelled R S should be labelled R S. p 213 l 10 3 12 1 Summary,should be 3 12 1 Terms Theorems and Algorithms. p 218 ll 1 7 Define the relation D on N so that n D m if and only if n m An. upper bound of two natural numbers in D is a natural number that both divide. The smallest such natural number is called the least upper bound and is denoted. as lub For example 6 is the least upper bound of 2 and 3 A lower bound of. two natural numbers in D is a naturally number that divides both numbers The. largest such natural number is called the greatest lower bound and is denoted. as glb For example the greatest lower bound of 4 and 6 is 2 Find. should be Let m n N m divides y denoted as m n if there exists an. integer j N such that m n n Let m n N such that m n An upper bound. of m and n is an integer j N such that m j and n j An upper bound always. exists as m n is an upper bound for m and n The smallest upper bound is. called the least upper bound denoted as lub m n Likewise a lower bound. of m and n is an integer j N such that j m and j n A lower bound always. exists as 1 is a lower bound for m and n The largest lower bound is called the. greatest lower bound denoted as glb m n For 4 and 6 glb 4 6 2 and. lub 4 6 12 Find,p 220 l 4 is seated at a chair,should be is seated in a chair. p 223 l 32 Example 4 1a the range of function, should be Example 1 a in Section 4 1 the range of function. p 225 Figure 4 2 The circles at the left end of the segments should be at the. right end of the segments, p 228 Figure 4 5 needs p q r changed to p q r as the third term. down in the middle output lines Also p q r changed to p q r in. the fifth term down in the middle output lines The final output shoul. computer switches in Section 1 3 4 and will be proved several times in the book using several di erent ideas replace with If you think of the elements of a set as switches that each can be set on or o the next result tells you how many di erent ways you can choose a subset of switches such that each element is on p 53 l 21 Remove The

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