Mathematical Proofs Transition to Advanced Mathematics Study Guide 2025/ 2026 Practice Problems with Solution

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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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manual forproofs
mathematical
a transition
proofs
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a transition
mathematics
to advanced
bymathematics
albert d polimeni
by albert
gary dchartrand
polimeni and
garyping.pdf
chartrand and ping

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Solution manual for mathematical
Solution
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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




Page 2 Solution manual for
Solution
mathematical
manualproofs
for mathematical
a transitionproofs
to advanced
a transition
mathematics
to advanced
by albert
mathematics
d polimeni
bygary
albert
chartrand
d polimeni
andgary
ping.pdf
chartrand and ping

, lOMoARcPSD|58847208




Solution manual for mathematical
Solution
proofs manual
a transition
for mathematical
to advanced mathematics
proofs
Page
a transition
3 of
by269
albert
to advanced
d polimenimathematics
gary chartrand
by albert
and ping.pdf
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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

Page 3 Solution manual for
Solution
mathematical
manualproofs
for mathematical
a transitionproofs
to advanced
a transition
mathematics
to advanced
by albert
mathematics
d polimeni
bygary
albert
chartrand
d polimeni
andgary
ping.pdf
chartrand and ping

, lOMoARcPSD|58847208




Solution manual for mathematical
Solution
proofs manual
a transition
for mathematical
to advanced mathematics
proofs
Page
a transition
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by269
albert
to advanced
d polimenimathematics
gary chartrand
by albert
and ping.pdf
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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



Page 4 Solution manual for
Solution
mathematical
manualproofs
for mathematical
a transitionproofs
to advanced
a transition
mathematics
to advanced
by albert
mathematics
d polimeni
bygary
albert
chartrand
d polimeni
andgary
ping.pdf
chartrand and ping