Solution Manual For Discrete Mathematics With Applications, 5th Edition

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Manual Section 1.11Chapter 1: Speaking MathematicallyMany college and university students have difficulty using and interpreting language involving if-thenstatements and quantification. Section 1.1 is a gentle introduction to the relation between informaland formal ways of expressing such statements. The exercises are intended to start the process ofhelping students improve their ability to interpret mathematical statements and evaluate their truthor 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 relatediscrete mathematics to the pre-calculus or calculus they have studied previously while enlargingtheir perspective to include a greater proportion of discrete examples.Section 1.4 is designed tobroaden students’ understanding of the way the word graph is used in mathematics and to showthem 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 andfunctions, such as transitivity and surjectivity, are considerably more complex than those used inChapter 4 to give students their first practice in constructing mathematical proofs. For this reasonset theory as a theory is left to Chapter 6, properties of functions to Chapter 7, and properties ofrelations to Chapter 8.By making slight changes about exercise choices, instructors could coverSection 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 theoremand its applications in Section 4.9. Instructors who wish to offer a self-contained treatment of graphtheory can combine both sections with the material in Chapter 10.College and university mathematics instructors may be surprised by the way students understandthe meaning of the term “real number.” When asked to evaluate the truth or falsity of a statementabout real numbers, it is not unusual for students to think only of integers.Thus an informaldescription of the relationship between real numbers and points on a number line is given in Section1.2 to illustrate that there are many real numbers between any pair of consecutive integers, Examples3.3.5 and 3.3.6 show that while there is a smallest positive integer there is no smallest positive realnumber, and the discussion in Chapter 7, which precedes the proof of the uncountability of the realnumbers between 0 and 1, describes a procedure for approximating the (possibly infinite) decimalexpansion for an arbitrarily chosen point on a number line.Section 1.11.a.x2=−1 (Or: the square ofxis−1)b. a real numberx2.a.a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6b.an integern;nis divided by 6 the remainder is 33.a.betweenaandbb. distinct real numbersaandb; there is a real numberc4.a.a real number; greater thanrb. real numberr; there is a real numbers5.a.ris positiveb.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)

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