Nguồn ngoài
Proofs and Concepts - The Fundamentals of Abstract Mathematics
This textbook provides an introduction to proofs, logic, sets, functions, and other fundamental topics of abstract mathematics. It is designed to be the textbook for a bridge course that introduces undergraduates to abstract mathematics, but it is also suitable for independent study by undergraduates (or mathematically mature high-school students), or for use as a very inexpensive supplement to undergraduate courses in any field of abstract mathematics.
- Tác giả
- Dave Witte Morris & Joy Morris
- Nhà xuất bản
- University of Lethbridge
- Nguồn
- LibreTexts
- Giấy phép
- CC BY-NC-SA
- Ngôn ngữ
- EN
- Danh mục
- Khoa học cơ bản
Mục lục· 73 mục
- 01: Propositional Logic
- 1.01: Assertions deductions and validity
- 1.02: Logic Puzzles
- 1.03: Using letters to symbolize assertions
- 1.04: Connectives
- 1.05: Determining whether an assertion is true
- 1.06: Tautologies and contradictions
- 1.07: Logical equivalence
- 1.08: Converse and contrapositive
- 1.09: Some valid deductions
- 1.10: Summary
- 02: Two-Column Proofs
- 2.01: First example of a two-column proof
- 2.02: Hypothesis an theorems in two-column proofs
- 2.03: Subproofs for introduction
- 2.04: Proof by contradiction
- 2.05: Proof strategies
- 2.06: What is a proof
- 2.07: Counterexamples
- 2.08: Summary
- 03: Sets
- 3.01: Propositional Logic is not enough
- 3.02: Sets subsets and predicates
- 3.03: Operations on sets
- 3.04: Summary
- 04: First-Order Logic
- 4.01: Quantifiers
- 4.02: Translating to First-Order logic
- 4.03: Negations
- 4.04: The Introduction and elimination rules for quantifiers
- 4.05: Some proofs about sets
- 4.06: Counterexamples (reprise)
- 4.07: Summary
- 05: Sample Topics
- 5.01: Number theory- divisibility and congruence
- 5.02: Abstract Algebra- commutative groups
- 5.03: Real analysis- convergent sequences
- 5.04: Summary
- 06: Functions
- 6.01: Cartesian product
- 6.02: Informal introduction to functions
- 6.03: Official Definition
- 6.04: One-to-one functions
- 6.05: Onto functions
- 6.06: Bijections
- 6.07: Inverse functions
- 6.08: Composition of functions
- 6.09: Image and pre-image
- 6.10: Summary
- 07: Equivalence relations
- 7.01: Binary relations
- 7.02: Definition and basic properties of equivalence relations
- 7.03: Equivalence classes
- 7.04: Modular arithmetic
- 7.05: Functions need to be well-defined
- 7.06: Partitions
- 7.07: Summary
- 08: Proof by Induction
- 8.01: The Principle of mathematical induction
- 8.02: Other proofs by induction
- 8.03: Other versions of induction
- 8.04: The natural numbers are well-ordered
- 8.05: Applications in number theory
- 8.06: Summary
- 09: Cardinality
- 9.01: Definition and basic properties
- 9.02: The pigeonhole principle
- 9.03: Cardinality of a union
- 9.04: Hotel infinity and the cardinality of infinite sets
- 9.05: Countable sets
- 9.06: uncountable sets
- 9.07: Summary