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What You Can Learn from Mathematical Proofs

A Mathematical Proofs Guide by the University of Dating​

"Mathematical Proofs: A Transition to Advanced Mathematics" is an introductory book on mathematical proofs, written by Gary Chartrand, Albert D. Polimeni, and Ping Zhang. It is aimed at first-year university students and learners of mathematics. This book serves as a bridge from computational mathematics to theoretical and abstract mathematics, providing a pathway for understanding and constructing mathematical proofs.

Here is a detailed overview of the book's contents:

1. Fundamental Concepts of Mathematical Proofs​

The book begins by explaining the importance and role of the concept of "proof." In mathematics, a proof is the logical process used to demonstrate that a statement or theorem is true. The authors emphasize the need to thoroughly understand mathematical logic, propositions, and the relationships between propositions in order to grasp proofs. They offer a clear explanation of the formal elements of proofs and their structure.

2. Logic and Propositions​

In order to perform proofs, it is necessary to first understand "propositions" and "logic." The book explains how propositions are formed and how they can be combined to create new propositions. A proposition is either true or false, and logical reasoning is used in the proof process. The book also covers basic operations on propositions, such as conjunction, disjunction, implication, and equivalence.

3. Introduction to Proof Techniques​

The book introduces various proof techniques, including the following:

  • Direct Proof: A method of proof where conclusions are directly derived from assumptions.
  • Proof by Contradiction: A method in which one assumes that a proposition is false and shows that this assumption leads to a contradiction, thereby proving the proposition to be true.
  • Mathematical Induction: A technique for proving propositions related to natural numbers, which involves a base case and an inductive step.
  • Counterexample: A method for proving that a proposition is false by providing a specific example where the proposition does not hold.

These proof techniques are explained in detail, with examples showing how they are applied to actual problems.

4. Set Theory and Functions​

Set theory is a foundational area of modern mathematics and is crucial when proving various mathematical statements. The authors explain the basic properties of sets (elements, subsets, intersection, union, difference, etc.) and introduce proof techniques related to sets. They also discuss the concept of functions and their properties, including proofs involving functions.

5. Properties of Real Numbers and Order​

The properties of real numbers, particularly related to order, upper and lower bounds, are critical in areas such as real analysis. Many proofs involving real numbers make use of their properties as an ordered field, making this section essential for a thorough understanding of real analysis.

6. Concrete Examples and Exercises​

In addition to theoretical content, the book provides numerous concrete examples and exercises, allowing readers to deepen their understanding through hands-on problem solving. The exercises range from simple problems to more challenging ones, helping readers practically apply the proof techniques they have learned.

7. Proof Style and Precision​

The book also addresses the importance of proof style and precision. It explains key considerations when writing proofs, such as logical consistency, clear explanations, and avoiding unnecessary verbosity. This section guides readers on how to write proofs correctly and efficiently.

8. The Importance of Mathematical Thinking​

The book aims not only to teach proof techniques but also to cultivate mathematical thinking. The authors emphasize the importance of logically organizing problems and progressing methodically toward proofs. This style of thinking is valuable not only in mathematics but also in solving problems in other fields.

Chapter Details​

"Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand, Albert D. Polimeni, and Ping Zhang is a book that systematically guides readers through the fundamental and advanced aspects of mathematical proofs. It covers essential proof techniques and concepts in mathematics, gradually developing theory through its chapters. Below is a detailed explanation of the contents of each chapter.

0. Communicating Mathematics​

This chapter emphasizes the communication skills needed for learning mathematics.

  • 0.1 Learning Mathematics: This section explains approaches and methods for learning mathematics effectively, stressing the importance of practice and repetition for deep understanding.

  • 0.2 What Others Have Said About Writing: Introduces perspectives on mathematical writing from other scholars, highlighting the need for logical and concise expression.

  • 0.3 Mathematical Writing: Provides essential guidelines for writing mathematical texts. It stresses the need for clarity and precision, avoiding ambiguity.

  • 0.4 Using Symbols: Discusses the use of symbols and equations. Mathematical symbols are an important tool for efficient communication while maintaining mathematical accuracy.

  • 0.5 Writing Mathematical Expressions: A detailed guide on how to write mathematical expressions correctly, including concrete examples.

  • 0.6 Common Words and Phrases in Mathematics: Introduces common terminology and phrases used in mathematics, which helps in understanding proofs and academic papers.

  • 0.7 Some Closing Comments About Writing: Summarizes key advice on writing good mathematical texts, offering tips for writing clear and effective mathematical papers.

1. Sets​

A chapter on set theory.

  • 1.1 Describing a Set: Definitions and methods for describing sets, which are collections of elements.

  • 1.2 Subsets: Explanation of the concept of subsets and their properties.

  • 1.3 Set Operations: Introduction to basic set operations like union, intersection, and difference.

  • 1.4 Indexed Collections of Sets: Focuses on indexed collections of sets, particularly how to handle infinite sets.

  • 1.5 Partitions of Sets: Discusses partitions of sets.

  • 1.6 Cartesian Products of Sets: Explains Cartesian products, or direct products, of two or more sets.

2. Logic​

A chapter on the foundations of logic.

  • 2.1 Statements: What are propositions (or statements) in mathematics? This section provides the basics of propositions.

  • 2.2 The Negation of a Statement: Explains the process of negating a proposition and its logical significance.

  • 2.3 The Disjunction and Conjunction of Statements: Discusses logical OR (disjunction) and logical AND (conjunction) and their properties.

  • 2.4 The Implication: Detailed explanation of the logical concept of implication (if-then statements).

  • 2.5 More On Implications: Further exploration of the properties and logical background of implications.

  • 2.6 The Biconditional: Explains biconditional statements (if and only if).

  • 2.7 Tautologies and Contradictions: Introduces tautologies (statements that are always true) and contradictions (statements that are always false).

  • 2.8 Logical Equivalence: Defines logical equivalence, where two propositions are logically equivalent.

  • 2.9 Some Fundamental Properties of Logical Equivalence: Explores the basic properties of logical equivalence.

  • 2.10 Quantified Statements: Discusses quantified propositions, such as universal and existential quantifiers.

  • 2.11 Characterizations of Statements: Characterizes propositions, particularly how to classify them as true or false.

3. Direct Proof and Proof by Contrapositive​

A chapter on direct proof and proof by contrapositive methods.

  • 3.1 Trivial and Vacuous Proofs: Introduces trivial proofs (obvious proofs) and vacuous proofs (proofs involving an empty set).

  • 3.2 Direct Proofs: A direct method of proof, where the conclusion is derived directly from the assumptions.

  • 3.3 Proof by Contrapositive: Proofs using the contrapositive of a statement.

  • 3.4 Proof by Cases: Explains proofs that are split into several cases.

  • 3.5 Proof Evaluations: Discusses how to evaluate the quality of a proposed proof.

4. More on Direct Proof and Proof by Contrapositive​

This chapter delves deeper into direct proofs and proofs by contrapositive.

  • 4.1 Proofs Involving Divisibility of Integers: Proofs related to the divisibility of integers.

  • 4.2 Proofs Involving Congruence of Integers: Proofs involving the concept of congruence in integers.

  • 4.3 Proofs Involving Real Numbers: Proofs related to real numbers.

  • 4.4 Proofs Involving Sets: Proofs involving sets.

  • 4.5 Fundamental Properties of Set Operations: Discusses fundamental properties of set operations.

  • 4.6 Proofs Involving Cartesian Products of Sets: Proofs related to Cartesian products of sets.

5. Existence and Proof by Contradiction​

A chapter on existence proofs and proof by contradiction.

  • 5.1 Counterexamples: The use and role of counterexamples in proofs.

  • 5.2 Proof by Contradiction: Explains the method of proof by contradiction.

  • 5.3 A Review of Three Proof Techniques: A review of the three proof techniques studied so far.

  • 5.4 Existence Proofs: The method of proving the existence of an object.

  • 5.5 Disproving Existence Statements: The process of disproving existence statements.

6. Mathematical Induction​

A chapter on mathematical induction.

  • 6.1 The Principle of Mathematical Induction: The basic principle of mathematical induction.

  • 6.2 A More General Principle of Mathematical Induction: A more generalized principle of mathematical induction.

  • 6.3 Proof By Minimum Counterexample: A proof technique using the minimum counterexample.

  • 6.4 The Strong Principle of Mathematical Induction: A stronger version of mathematical induction.

7. Reviewing Proof Techniques​

This chapter reviews and deepens the understanding of the proof techniques learned so far.

  • 7.1 Reviewing Direct Proof and Proof by Contrapositive: A review of direct proof and proof by contrapositive.

  • 7.2 Reviewing Proof by Contradiction and Existence Proofs: A review of proof by contradiction and existence proofs.

  • 7.3 Reviewing Induction Proofs: A review of proofs by induction.

  • 7.4 Reviewing Evaluations of Proposed Proofs: A review of how to evaluate proposed proofs.

8. Prove or Disprove​

This chapter focuses on methods to prove or disprove a proposition.

  • 8.1 Conjectures in Mathematics: Discusses the importance of conjectures in mathematics and how to prove or disprove them.

  • 8.2 Revisiting Quantified Statements: Revisits quantified statements and how to prove or disprove them.

  • 8.3 Testing Statements: Methods for testing statements to determine their truth value.

9. Equivalence Relations​

A chapter on equivalence relations.

  • 9.1 Relations: The definition of relations and understanding relationships between elements in sets.

  • 9.2 Properties of Relations: Basic properties of relations, including reflexivity, symmetry, and transitivity.

  • 9.3 Equivalence Relations: Detailed explanation of equivalence relations and their properties.

  • 9.4 Properties of Equivalence Classes: Discusses the properties of equivalence classes and partitioning using equivalence relations.

  • 9.5 Congruence Modulo n: Explains congruence modulo nn and its significance in mathematics.

  • 9.6 The Integers Modulo n: Discusses the concept of integers modulo nn, with a focus on modular arithmetic.

10. Functions​

A chapter on the basics of functions and their properties.

  • 10.1 The Definition of Function: The definition of a function and its basic properties.

  • 10.2 The Set of All Functions From A to B: Discusses the set of all functions from set AA to set BB.

  • 10.3 One-to-one and Onto Functions: Defines and explains one-to-one (injective) and onto (surjective) functions.

  • 10.4 Bijective Functions: Discusses bijective functions, which are both one-to-one and onto.

  • 10.5 Composition of Functions: The composition of functions and its properties.

  • 10.6 Inverse Functions: Defines inverse functions and discusses how to find them.

  • 10.7 Permutations: Explains the concept of permutations, including calculations and properties.

11. Cardinalities of Sets​

A chapter on the cardinality (size) of sets.

  • 11.1 Numerically Equivalent Sets: Discusses what it means for sets to be numerically equivalent, including definitions and examples.

  • 11.2 Denumerable Sets: The concept of countable sets and how they can be enumerated, even if infinite.

  • 11.3 Uncountable Sets: Discusses uncountable sets, sets that are too large to be counted, even infinitely.

  • 11.4 Comparing Cardinalities of Sets: Methods for comparing the cardinalities (sizes) of sets.

  • **11.5 The

Continuum Hypothesis**: Introduces the continuum hypothesis, a famous problem in set theory.

12. Relations​

This chapter provides a detailed explanation of relations.

  • 12.1 Relations: A basic definition and properties of relations are introduced. A relation is a connection between elements of a set.

  • 12.2 Properties of Relations: This section covers important properties of relations, such as reflexivity (self-relationship), symmetry (inverse relationships), and transitivity (the chain-like property of relations).

  • 12.3 Partial Order: This section explains partial orders, a type of relation that defines a "partial" order in a set.

  • 12.4 Total Order: Total order relations are discussed, where every pair of elements is comparable.

  • 12.5 Equivalence Relations and Partitions: This part discusses equivalence relations and how they partition a set into equivalence classes.

13. Graphs and Trees​

In this chapter, graph theory and tree structures are explored.

  • 13.1 Graphs: The basic definition of graphs, their components (vertices and edges), and types of graphs (directed, undirected) are introduced.

  • 13.2 Types of Graphs: Various types of graphs, including complete graphs, subgraphs, and bipartite graphs, are described.

  • 13.3 Trees: This section covers trees, which are connected graphs without cycles. The definition and properties of trees are examined.

  • 13.4 Spanning Trees: Spanning trees are discussed. A spanning tree is a minimal tree that contains all the vertices of a graph.

  • 13.5 Graph Algorithms: Basic graph algorithms, such as Depth-First Search (DFS) and Breadth-First Search (BFS), are introduced.

14. Counting and Combinatorics​

This chapter introduces the techniques of combinatorics and counting.

  • 14.1 Permutations: This section explains permutations, which refer to different arrangements of a set of objects where order matters.

  • 14.2 Combinations: Combinations are explored here, where order does not matter in selecting objects from a set.

  • 14.3 The Binomial Theorem: The binomial theorem is introduced, explaining how to expand powers of binomials.

  • 14.4 Pigeonhole Principle: The pigeonhole principle is explained, which states that if more objects are placed into fewer containers than there are objects, at least one container must contain more than one object.

  • 14.5 Inclusion-Exclusion Principle: The principle of inclusion and exclusion is used to calculate the size of the union of several sets.

15. Introduction to Number Theory​

This chapter covers the basics of number theory.

  • 15.1 Divisibility: The concept of divisibility in integers is discussed, exploring how integers divide each other.

  • 15.2 Prime Numbers: The definition and importance of prime numbers are introduced, along with their properties and role in factorization.

  • 15.3 The Fundamental Theorem of Arithmetic: The fundamental theorem of arithmetic is discussed, stating that every integer greater than 1 can be uniquely factored into primes.

  • 15.4 Greatest Common Divisor (GCD): The greatest common divisor and methods for calculating it (e.g., Euclidean algorithm) are covered.

  • 15.5 Least Common Multiple (LCM): The least common multiple of integers and its calculation is introduced.

16. Recurrence Relations​

This chapter introduces recurrence relations.

  • 16.1 Recurrence Relations: A recurrence relation is a way of defining a sequence of numbers using previous terms. This section covers various types of recurrence relations.

  • 16.2 Solving Recurrence Relations: Methods for solving recurrence relations, especially linear recurrence relations, are introduced.

  • 16.3 Generating Functions: Generating functions are explored as a tool to solve recurrence relations.

  • 16.4 Fibonacci Numbers: The Fibonacci sequence is introduced as an example of a recurrence relation, demonstrating its applications.

17. Linear Algebra​

This chapter introduces the basics of linear algebra.

  • 17.1 Vectors and Matrices: Definitions and operations related to vectors and matrices are discussed.

  • 17.2 Systems of Linear Equations: This section covers systems of linear equations and methods for solving them, such as Gaussian elimination.

  • 17.3 Matrix Operations: Various matrix operations, including addition, multiplication, and finding the inverse, are explored.

  • 17.4 Determinants: The determinant of a matrix is defined and methods for calculating it are explained.

  • 17.5 Eigenvalues and Eigenvectors: The concepts of eigenvalues and eigenvectors are introduced, along with methods for diagonalizing matrices.

18. Applications of Proofs​

This chapter focuses on real-world applications of proofs.

  • 18.1 Proofs in Geometry: The application of proof techniques in geometry is discussed, including methods for proving geometric theorems.

  • 18.2 Proofs in Algebra: Proof techniques in algebra, particularly for proving properties of algebraic structures, are examined.

  • 18.3 Proofs in Number Theory: The chapter discusses proof techniques used in number theory.

  • 18.4 Proofs in Computer Science: The application of proof techniques in computer science, particularly in algorithms and complexity theory, is covered.

19. Conclusion​

The final chapter summarizes the key concepts and offers guidance for future study.

  • 19.1 Review of Proof Techniques: A review of the various proof techniques covered throughout the book and the contexts in which they are most effective.

  • 19.2 Further Reading and Study: Recommendations for further reading and study materials to deepen understanding.

  • 19.3 Final Thoughts: Final reflections on the importance of mathematical proofs and encouragement to continue learning.

Conclusion​

Mathematical Proofs: A Transition to Advanced Mathematics covers the fundamentals to advanced applications of mathematical proofs, making it an excellent introduction for beginners to step into the world of abstract mathematics. By understanding proof techniques, one can develop the ability to handle mathematical logic, laying the foundation for further studies in more advanced areas of mathematics.

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