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Albert D. Polimeni, Gary
Chartrand, Ping Zhang - Solution
Manual for Mathematical Proofs
A Transition to
Advanced Mathematics
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Mathematical Proofs
A Transition to
Advanced Mathematics
Fourth Edition
Gary Chartrand
Western Michigan University
Albert D. Polimeni
State University of New York at Fredonia
Ping Zhang
Western Michigan University
oofs
and aping.pdfY!Solution
transition
Solutionto manual
advanced
manual
for mathematics
mathematical
for mathematical
by
proofs
albert
proofs
adtransition
polimeni
a transition
to
gary
advanced
to
chartrand
advanced
mathematics
and
mathematics
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
by albert
by albert
d polimeni
d polimeni
gary chartrand
gary chartrand
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advanced
for mathematics
mathematical
manualbyproofs
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mathematical
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polimeniproofs
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gary
advanced
chartrand
a transition
mathematics
and
to advanced
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
by albert
mathematics
d polimeni
by gary
albertchartrand
d polimeni
and
gary
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS
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Table of Contents
0. Communicating Mathematics
0.1 Learning Mathematics
0.2 What Others Have Said About Writing
0.3 Mathematical Writing
0.4 Using Symbols
0.5 Writing Mathematical Expressions
0.6 Common Words and Phrases in Mathematics
0.7 Some Closing Comments About Writing
1. Sets
1.1 Describing a Set
1.2 Subsets
1.3 Set Operations
1.4 Indexed Collections of Sets
1.5 Partitions of Sets
1.6 Cartesian Products of Sets Exercises for Chapter 1
2. Logic
2.1 Statements
2.2 Negations
2.3 Disjunctions and Conjunctions
2.4 Implications
2.5 More on Implications
2.6 Biconditionals
2.7 Tautologies and Contradictions
2.8 Logical Equivalence
2.9 Some Fundamental Properties of Logical Equivalence
2.10 Quantified Statements
2.11 Characterizations Exercises for Chapter 2
3. Direct Proof and Proof by Contrapositive
3.1 Trivial and Vacuous Proofs
3.2 Direct Proofs
3.3 Proof by Contrapositive
3.4 Proof by Cases
3.5 Proof Evaluations
Exercises for Chapter 3
4. More on Direct Proof and Proof by Contrapositive
4.1 Proofs Involving Divisibility of Integers
4.2 Proofs Involving Congruence of Integers
4.3 Proofs Involving Real Numbers
4.4 Proofs Involving Sets
4.5 Fundamental Properties of Set Operations
4.6 Proofs Involving Cartesian Products of Sets Exercises for Chapter 4
5. Existence and Proof by Contradiction
5.1 Counterexamples
5.2 Proof by Contradiction
iv
5.3 A Review of Three Proof Techniques
oofs
and aping.pdfY!Solution
transition
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advanced
manual
for mathematics
mathematical
for mathematical
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proofs
albert
proofs
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gary
advanced
to
chartrand
advanced
mathematics
and
mathematics
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
by albert
by albert
d polimeni
d polimeni
gary chartrand
gary chartrand
and ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS
and ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GES
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advanced
for mathematics
mathematical
manualbyproofs
foralbert
mathematical
adtransition
polimeniproofs
to
gary
advanced
chartrand
a transition
mathematics
and
to advanced
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
by albert
mathematics
d polimeni
by gary
albertchartrand
d polimeni
and
gary
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS
chartrand and ping.pdfY!G*EIH(*#Y!(*#Y#*
5.4 Existence Proofs
5.5 Disproving Existence Statements Exercises for Chapter 5
6. Mathematical Induction
6.1 The Principle of Mathematical Induction
6.2 A More General Principle of Mathematical Induction
6.3 The Strong Principle of Mathematical Induction
6.4 Proof by Minimum Counterexample Exercises for Chapter 6
7. Reviewing Proof Techniques
7.1 Reviewing Direct Proof and Proof by Contrapositive
7.2 Reviewing Proof by Contradiction and Existence Proofs
7.3 Reviewing Induction Proofs
7.4 Reviewing Evaluations of Proposed Proofs Exercises for Chapter 7
8. Prove or Disprove
8.1 Conjectures in Mathematics
8.2 Revisiting Quantified Statements
8.3 Testing Statements Exercises for Chapter 8
9. Equivalence Relations
9.1 Relations
9.2 Properties of Relations
9.3 Equivalence Relations
9.4 Properties of Equivalence Classes
9.5 Congruence Modulo n
9.6 The Integers Modulo n Exercises for Chapter 9
10. Functions
10.1 The Definition of Function
10.2 One-to-one and Onto Functions
10.3 Bijective Functions
10.4 Composition of Functions
10.5 Inverse Functions
Exercises for Chapter 10
11. Cardinalities of Sets
11.1 Numerically Equivalent Sets
11.2 Denumerable Sets
11.3 Uncountable Sets
11.4 Comparing Cardinalities of Sets
11.5 The Schroder-Bernstein Theorem¨ Exercises for Chapter 11
12. Proofs in Number Theory
12.1 Divisibility Properties of Integers
12.2 The Division Algorithm
12.3 Greatest Common Divisors
v
12.4 The Euclidean Algorithm
12.5 Relatively Prime Integers
12.6 The Fundamental Theorem of Arithmetic
12.7 Concepts Involving Sums of Divisors Exercises for Chapter 12
oofs
and aping.pdfY!Solution
transition
Solutionto manual
advanced
manual
for mathematics
mathematical
for mathematical
by
proofs
albert
proofs
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polimeni
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gary
advanced
to
chartrand
advanced
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and
mathematics
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
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gary chartrand
gary chartrand
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and ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GES
ping.pdfY!G*EIH(*#Y!(Solution
Solution
to advanced
manual
mathematics
for mathematical
manual
by albert
forproofs
mathematical
d polimeni
a transition
gary
proofs
tochartrand
advanced
a transition
andmathematics
to
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
advanced
bymathematics
albert d polimeni
by albert
garydchartrand
polimeni and
garyping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GES
chartrand and ping.pdfY!G*EIH(*#Y!(*#Y
Albert D. Polimeni, Gary
Chartrand, Ping Zhang - Solution
Manual for Mathematical Proofs
A Transition to
Advanced Mathematics
adtransition
ping.pdfY!Solution
Solution
to advanced
manual
manual
mathematics
for for
mathematical
mathematical
by albert
proofs
proofs
d polimeni
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a transition
garytochartrand
advanced
to advanced
and
mathematics
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
mathematics
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by albert
d polimeni
d polimeni
gary
gary
chartrand
chartrand
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ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*G
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advanced
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mathematical
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gary
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chartrand
a transition
mathematics
and
to advanced
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
by albert
mathematics
d polimeni
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chartrand and ping.pdfY!G*EIH(*#Y!(*#Y#*
Mathematical Proofs
A Transition to
Advanced Mathematics
Fourth Edition
Gary Chartrand
Western Michigan University
Albert D. Polimeni
State University of New York at Fredonia
Ping Zhang
Western Michigan University
oofs
and aping.pdfY!Solution
transition
Solutionto manual
advanced
manual
for mathematics
mathematical
for mathematical
by
proofs
albert
proofs
adtransition
polimeni
a transition
to
gary
advanced
to
chartrand
advanced
mathematics
and
mathematics
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
by albert
by albert
d polimeni
d polimeni
gary chartrand
gary chartrand
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and ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GES
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advanced
for mathematics
mathematical
manualbyproofs
foralbert
mathematical
adtransition
polimeniproofs
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gary
advanced
chartrand
a transition
mathematics
and
to advanced
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
by albert
mathematics
d polimeni
by gary
albertchartrand
d polimeni
and
gary
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS
chartrand and ping.pdfY!G*EIH(*#Y!(*#Y#*
Table of Contents
0. Communicating Mathematics
0.1 Learning Mathematics
0.2 What Others Have Said About Writing
0.3 Mathematical Writing
0.4 Using Symbols
0.5 Writing Mathematical Expressions
0.6 Common Words and Phrases in Mathematics
0.7 Some Closing Comments About Writing
1. Sets
1.1 Describing a Set
1.2 Subsets
1.3 Set Operations
1.4 Indexed Collections of Sets
1.5 Partitions of Sets
1.6 Cartesian Products of Sets Exercises for Chapter 1
2. Logic
2.1 Statements
2.2 Negations
2.3 Disjunctions and Conjunctions
2.4 Implications
2.5 More on Implications
2.6 Biconditionals
2.7 Tautologies and Contradictions
2.8 Logical Equivalence
2.9 Some Fundamental Properties of Logical Equivalence
2.10 Quantified Statements
2.11 Characterizations Exercises for Chapter 2
3. Direct Proof and Proof by Contrapositive
3.1 Trivial and Vacuous Proofs
3.2 Direct Proofs
3.3 Proof by Contrapositive
3.4 Proof by Cases
3.5 Proof Evaluations
Exercises for Chapter 3
4. More on Direct Proof and Proof by Contrapositive
4.1 Proofs Involving Divisibility of Integers
4.2 Proofs Involving Congruence of Integers
4.3 Proofs Involving Real Numbers
4.4 Proofs Involving Sets
4.5 Fundamental Properties of Set Operations
4.6 Proofs Involving Cartesian Products of Sets Exercises for Chapter 4
5. Existence and Proof by Contradiction
5.1 Counterexamples
5.2 Proof by Contradiction
iv
5.3 A Review of Three Proof Techniques
oofs
and aping.pdfY!Solution
transition
Solutionto manual
advanced
manual
for mathematics
mathematical
for mathematical
by
proofs
albert
proofs
adtransition
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gary
advanced
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chartrand
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and
mathematics
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
by albert
by albert
d polimeni
d polimeni
gary chartrand
gary chartrand
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transition
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advanced
for mathematics
mathematical
manualbyproofs
foralbert
mathematical
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polimeniproofs
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gary
advanced
chartrand
a transition
mathematics
and
to advanced
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
by albert
mathematics
d polimeni
by gary
albertchartrand
d polimeni
and
gary
ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS
chartrand and ping.pdfY!G*EIH(*#Y!(*#Y#*
5.4 Existence Proofs
5.5 Disproving Existence Statements Exercises for Chapter 5
6. Mathematical Induction
6.1 The Principle of Mathematical Induction
6.2 A More General Principle of Mathematical Induction
6.3 The Strong Principle of Mathematical Induction
6.4 Proof by Minimum Counterexample Exercises for Chapter 6
7. Reviewing Proof Techniques
7.1 Reviewing Direct Proof and Proof by Contrapositive
7.2 Reviewing Proof by Contradiction and Existence Proofs
7.3 Reviewing Induction Proofs
7.4 Reviewing Evaluations of Proposed Proofs Exercises for Chapter 7
8. Prove or Disprove
8.1 Conjectures in Mathematics
8.2 Revisiting Quantified Statements
8.3 Testing Statements Exercises for Chapter 8
9. Equivalence Relations
9.1 Relations
9.2 Properties of Relations
9.3 Equivalence Relations
9.4 Properties of Equivalence Classes
9.5 Congruence Modulo n
9.6 The Integers Modulo n Exercises for Chapter 9
10. Functions
10.1 The Definition of Function
10.2 One-to-one and Onto Functions
10.3 Bijective Functions
10.4 Composition of Functions
10.5 Inverse Functions
Exercises for Chapter 10
11. Cardinalities of Sets
11.1 Numerically Equivalent Sets
11.2 Denumerable Sets
11.3 Uncountable Sets
11.4 Comparing Cardinalities of Sets
11.5 The Schroder-Bernstein Theorem¨ Exercises for Chapter 11
12. Proofs in Number Theory
12.1 Divisibility Properties of Integers
12.2 The Division Algorithm
12.3 Greatest Common Divisors
v
12.4 The Euclidean Algorithm
12.5 Relatively Prime Integers
12.6 The Fundamental Theorem of Arithmetic
12.7 Concepts Involving Sums of Divisors Exercises for Chapter 12
oofs
and aping.pdfY!Solution
transition
Solutionto manual
advanced
manual
for mathematics
mathematical
for mathematical
by
proofs
albert
proofs
adtransition
polimeni
a transition
to
gary
advanced
to
chartrand
advanced
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and
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ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS*(@U(*UE@&*EH*HJE@&EH*@(o8
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gary chartrand
gary chartrand
and ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GESMSS
and ping.pdfY!G*EIH(*#Y!(*#Y#*!#!&**#*GES
