and graphs. Including Sections 1.2 and 1.3 at the beginning of the course can help students relate

Instructor's Manual Section 1.11

Chapter 1: Speaking Mathematically

Many college and university students have difficulty using and interpreting language involving if-then statements and quantification. Section 1.1 is a gentle introduction to the relation between informal and formal ways of expressing such statements. The exercises are intended to start the process of helping students improve their ability to interpret mathematical statements and evaluate their truth or falsity. Sections 1.2 - 1.4 are a brief introduction to the language of sets, relations, functions, and graphs. Including Sections 1.2 and 1.3 at the beginning of the course can help students relate discrete mathematics to the pre-calculus or calculus they have studied previously while enlarging their perspective to include a greater proportion of discrete examples. Section 1.4 is designed to broaden students’ understanding of the way the word graph is used in mathematics and to show them how graph models can be used to solve some significant problems.Proofs of set properties, such as the distributive laws, and proofs of properties of relations and functions, such as transitivity and surjectivity, are considerably more complex than those used in Chapter 4 to give students their first practice in constructing mathematical proofs. For this reason set theory as a theory is left to Chapter 6, properties of functions to Chapter 7, and properties of relations to Chapter 8. By making slight changes about exercise choices, instructors could cover Section 1.2 just before starting Chapter 6 and Section 1.3 just before starting Chapter 7.The material in Section 1.4 lays the groundwork for the discussion of the handshake theorem and its applications in Section 4.9. Instructors who wish to offer a self-contained treatment of graph theory can combine both sections with the material in Chapter 10.College and university mathematics instructors may be surprised by the way students understand the meaning of the term eal number.” When asked to evaluate the truth or falsity of a statement about real numbers, it is not unusual for students to think only of integers. Thus an informal description of the relationship between real numbers and points on a number line is given in Section 1.2 to illustrate that there are many real numbers between any pair of consecutive integers, Examples

3.3.5 and 3.3.6 show that while there is a smallest positive integer there is no smallest positive real

number, and the discussion in Chapter 7, which precedes the proof of the uncountability of the real numbers between 0 and 1, describes a procedure for approximating the (possibly infinite) decimal expansion for an arbitrarily chosen point on a number line.Section 1.1 1.a.x 2

=−1 (Or: the square ofxis−1)b. a real numberx

2.a.a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6 b.an integern;nis divided by 6 the remainder is 3 3.a.betweenaandbb. distinct real numbersaandb; there is a real numberc 4.a.a real number; greater thanr b. real numberr; there is a real numbers 5.a.ris positive

b.positive; the reciprocal ofris positive (Or: positive; 1/ris positive)

c.is positive; 1/ris positive (Or: is positive; the reciprocal ofris positive)

6.a.sis negativeb.negative; the cube root ofsis negative (Or:

3 √ sis negative) c.is negative; 3 √

sis negative (Or: the cube root ofsis negative)

(Discrete Mathematics with Applications, 5e Susanna Epp) (Solution Manual all Chapters) 1 / 4

2 Instructor's Manual: Chapter 1

7.a.There are real numbers whose sum is less than their difference. True. For example,

• + (−1) = 0;1−(−1) = 1 + 1 = 2, and 0<2.

b.There is a real number whose square is less than itself. True. For example, (1=2) 2 = 1=4< 1=2 .c.The square of each positive integer is greater than or equal to the integer.True. Ifnis any positive integer, thenn≥1. Multiplying both sides by the positive number ndoes not change the direction of the inequality (see Appendix A, T20), and son 2 ≥n.d.The absolute value of the sum of any two numbers is less than or equal to the sum of their absolute values.True. This is known as the triangle inequality. It is discussed in Section 4.4.

8.a.have four sidesb.has four sidesc.has four sidesd.is a square; has four sides e.Jhas four sides 9.a.have at most two real solutionsb.has at most two real solutionsc.has at most two real solutionsd.is a quadratic equation; has at most two real solutionse.Ehas at most two real solutions 10.a.have reciprocalsb.a reciprocalc.sis a reciprocal forr 11.a.have positive square rootsb.a positive square rootc.ris a square root fore 12.a.real number; product with every number leaves the number unchanged b.a positive square rootc.rs=s 13.a.real number; product with every real number equals zero b.with every number leaves the number unchangedc.ab= 0 Section 1.2 1.A=CandB=D 2.a.The set of all positive real numbersxsuch that 0 is less thanxandxis less than 1 b.The set of all real numbersxsuch thatxis less than or equal to zero orxis greater than or equal to 1 c.The set of all integersnsuch thatnis a factor of 6 d.The set of all positive integersnsuch thatnis a factor of 6 3.a.No,{4}is a set with one element, namely 4, whereas 4 is just a symbol that represents the number 4

b.Three: the elements of the set are 3, 4, and 5.

c.Three: the elements are the symbol 1, the set{1}, and the set{1;{1}}

4.a.Yes:{2}is the set whose only element is 2.b.One: 2 is the only element in this setc.Two: The two elements are 0 and{0}d.Yes:{0}is one of the elements listed in the set.e.No: The only elements listed in the set are{0}and{1}, and 0 is not equal to either of these. 2 / 4

Instructor's Manual Section 1.23

5.The only sets that are equal to each other areAandD:

Acontains the integers 0, 1, and 2 and nothing else.Bcontains all the real numbers that are greater than or equal to−1 and less than 3.Ccontains all the real numbers that are greater than−1 and less than 3. Thus−1 is inB but not inC.Dcontains all the integers greater than−1 and less than 3. ThusDcontains the integers 0, 1, and 2 and nothing else, and soD={0;1;2}=A.Econtains all the positive integers greater than−1 and less than 3. HenceEcontains the integers 1 and 2 and nothing else, that is,E={1;2}.

6.T2andT−3each have two elements, andT0andT1each have one element.

Justication:T2={2;2

2

}={2;4},T−3={−3;(−3)

2

}={−3;9},

T1={1;1

2 }={1;1}={1}, andT0={0;0 2

}={0;0}={0}.

7.a.{1;−1} b.T={m∈Z|m= 1+(−1) k

for some integerk}={0;2}:Exercises in Chapter 4 explore the

fact that (−1) k =−1 whenkis odd and (−1) k = 1 whenkis even. So 1+(−1) k

= 1+(−1) = 0

whenkis odd, and 1 + (−1) k = 1 + 1 = 2 whenkis even.c.the set has no elements

• Z(every integer is in the set)

e.There are no elements inWbecause there are no integers that are both greater than 1 and

less than−3:

f.X=Zbecause every integerusatisfies at least one of the conditionsu≤4 oru≥1: 8.a.No,B*Abecausej∈Bandj =∈A b.Yes, because every element inCis inA.c.Yes, because every element inCis inC.c.. Yes, because it is true that every element inCis inC.d.Yes,Cis a proper subset ofA. Both elements ofCare inA, butAcontains elements (namelycandf) that are not inC.

9.a.Yes b.No, the number 1 is not a set and so it cannot be a subset.c.No: The only elements in{1;2}are 1 and 2;and{2}is not equal to either of these.

d.Yes:{3}is one of the elements listed in{1;{2};{3}}.

e.Yes:{1}is the set whose only element is 1.

f.No, the only element in{2}is the number 2 and the number 2 is not one of the three elements in{1;{2};{3}}.

g.Yes: The only element in{1}is 1, and 1 is an element in{1;2}.

h.No: The only elements in{{1};2}are{1}and 2, and 1 is not equal to either of these.

i.Yes, the only element in{1}is the number 1, which is an element in{1;{2}}:

j.Yes: The only element in{1}is 1, which is is an element in{1}. So every element in{1}is in{1}.

10.a.No. Observe that (−2) 2 = (−2)(−2) = 4, whereas−2 2

=−(2

2

• =−4. So ((−2)

2 ;−2 2

• =

(4;−4), whereas (−2 2

;(−2)

2

• = (−4;4). And (4;−4)̸= (−4;4) because−4̸= 4.

b.No: For two ordered pairs to be equal, the elements in each pair must occur in the same order. In this case the first element of the first pair is 5, whereas the first element of the second 3 / 4

4 Instructor's Manual: Chapter 1

pair is−5, and the second element of the first pair is−5 whereas the second element of the second pair is 5.c.Yes. Note that 8−9 =−1 and 3 √ −1 =−1, and so (8−9; 3 √

−1) = (−1;−1).

d.Yes The first elements of both pairs equal 1 2 , and the second elements of both pairs equal −8.

11.a.{(w; a);(w; b);(x; a);(x; b);(y; a);(y; b);(z; a);(z; b)}A×Bhas 4·2 = 8 elements.b.{(a; w);(b; w);(a; x);(b; x);(a; y);(b; y);(a; z);(b; z)}B×Ahas 4·2 = 8 elements.c.{(w; w);(w; x);(w; y);(w; z);(x; w);(x; x);(x; y);(x; z);(y; w);(y; x);(y; y); (y; z);(z; w);(z; x);(z; y);(z; z)}A×Ahas 4·4 = 16 elements.d.{(a; a);(a; b);(b; a);(b; b)}B×Bhas 2·2 = 4 elements.

12.All four sets have nine elements.a.S×T={(2;1);(2;3);(2;5);(4;1);(4;3);(4;5);(6;1);(6;3);(6;5)} b.T×S={(1;2);(3;2);(5;2);(1;4);(3;4);(5;4);(1;6);(3;6);(5;6)} c.S×S={(2;2);(2;4);(2;6);(4;2);(4;4);(4;6);(6;2);(6;4);(6;6)} d.T×T={(1;1);(1;3);(1;5);(3;1);(3;3);(3;5);(5;1);(5;3);(5;5)} 13.a.A×(B×C) ={(1;(u; m));(1;(u; n));(2;(u; m));(2;(u; n));(3;(u; m));(3;(u; n))} b.(A×B)×C={((1; u); m);((1; u); n);((2; u); m);((2; u); n);((3; u); m);((3; u); n)} c.A×B×C={(1; u; m);(1; u; n);(2; u; m);(2; u; n);(3; u; m);(3; u; n)} 14.a.R×(S×T) ={(a;(x; p));(a;(x; q));(a;(x; r));(a;(y; p));(a;(y; q));(a;(y; r))} b.(R×S)×T={((a; x); p);((a; x); q);((a; x); r);((a; y); p);((a; y); q);((a; y); r)} c.R×S×T={(a; x; p);(a; x; q);(a; x; r);(a; y; p);(a; y; q);(a; y; r)}

15.0000, 0001, 0010, 0100, 1000

16.yxxxx; xyxxx,xxyxx,xxxyx,xxxxy Section 1.3 1.a.No. Yes. No. Yes.b.R={(2;6);(2;8);(2;10);(3;6);(4;8)} c.Domain ofR=A={2;3;4}, co-domain ofR=B={6;8;10} d.R 6 8 10 2 3 4 2.a.2S2 because 1 2 − 1 2 = 0, which is an integer.−1S−1 because 1 −1 − 1 −1 = 0, which is an integer.3S3 because 1 3 − 1 3 = 0, which is an integer.3/S−3 because 1 3 − 1 −3 = 2 3 , which is not an integer.

• / 4