GUIDE TO REVIEW QUESTIONS FOR CHAPTER 1
1.
2.
a) See p. 3. b) This is not a boring course.
a) See pp. 4, 5, 6, and 9.
b) Disjunction: rlll go to the movies tonight or Ill finish my discrete mathematics homework. Conjunction:
Ill go to the movies tonight a

SUPPLEMENTARY EXERCISES FOR CHAPTER 1
2. The truth table is as follows.
(P V q) - (p A or)
":3
meat36666 <
Q
pAn
F
T
F
T
F
F
F
F
mmmmHriae
"U'r-il-EE'TJHHQ
mamaweqe
Hr-HHZITJH'TJH'TJ
4. a) The converse is Ifl drive to work today, then it will rain. The

3 8
Chapter 1 The Foundations: Logic and Proofs
SECTION 1.7 Proof Methods and Strategy
11.
The preamble to the solutions for Section 1.6 applies here as well, so you might want to reread it at this time.
In addition, the section near the back of this Guid

SECTION 1.5 Rules of Inference
2.
This is modus tollens. The rst statement is p A q, where p is George does not have eight legs and q is
George is not an insect. The second statement is mq. The third is op. Modus tollens is valid. We can
therefore conclud

34
Chapter l The Foundations: Logic and Proofs
SECTION 1.6 Introduction to Proofs
This introduction applies jointly to this section and the next (1.? .
Learning to construct good mathematical proofs takes years. There is no algorithm for constructing the

26
Chapter 1 The Foundations: Logic and Proofs
SECTION 1.6 Introduction to Proofs
2.
10.
12.
14.
16.
18.
20.
We must show that whenever we have two even integers, their sum is even. Suppose that a and 1') are
two even integers. Then there exist integers s

Section 1.1
Propositional Logic 1
CHAPTER 1
The Foundations: Logic and Proofs
SECTION 1.1 Propositional Logic
ivlanipulating propositions and constructing truth tables are straightforward. A truth table is constructed by
finding the truth values of compou

SECTION 1.4 Nested Quantiers
Nested quantiers are one of the most difcult things for students to understand. The theoretical denition of
limit in calculus, for example, is hard to comprehend because it has three levels of nested quantiers. Study
the examp

10
Chapter 1 The Foundations: Logic and Proofs
SECTION 1.2 Propositional Equivalences
The solutions to Exercises llO are routine; we use truth tables to show that a proposition is a tautology or
that two propositions are equivalent. The reader should do m

12
Chapter 1 The Foundations: Logic and Proofs
SECTION 1.3 Predicates and Quantiers
2.
10.
12.
a) This is true, since there is an a. in orange. b) This is false, since there is no a in lemon.
c) This is false, since there is no a in true. (:1) This is tru

Section 1.1
Propositional Logic 1
CHAPTER 1
The Foundations: Logic and Proofs
SECTION 1.1 Propositional Logic
2.
Propositions must have clearly dened truth values, so a proposition must be a declarative sentence with no
free variables.
a) This is not a pr

SECTION 1.2 Propositional Equivalences
2. There are two cases. If p is true, then n(-up) is the negation of a false proposition, hence true. Similarly if p
is false, then I(np) is also false. Therefore the two propositions are logically equivalent.
4. a)