Central Library, Indian Institute of Technology Delhi
केंद्रीय पुस्तकालय, भारतीय प्रौद्योगिकी संस्थान दिल्ली

An introduction to proofs with set theory / Daniel Ashlock, Colin Lee.

By: Ashlock, Daniel [author.]Contributor(s): Lee, Colin R [author.]Material type: TextTextSeries: Synthesis digital library of engineering and computer science | Synthesis lectures on mathematics and statistics ; #35.Publisher: [San Rafael, California] : Morgan & Claypool, [2020]Description: 1 PDF (xv, 233 pages) : illustrations (some color)Content type: text Media type: electronic Carrier type: online resourceISBN: 9781681738802Subject(s): Set theory | Proof theory | set theory | proof | mathematical proofs | logic | functions | relations | integers | cardinal numbers | ordinal numbers | enumerative combinatoricsAdditional physical formats: Print version:: No titleDDC classification: 511.3/22 LOC classification: QA248 | .A755 2020ebOnline resources: Abstract with links to full text | Abstract with links to resource Also available in print.
Contents:
1. Introduction and review of background material -- 1.1. Polynomials and series -- 1.2. Representing numbers -- 1.3. Simple counting -- 1.4. Problems
2. Boolean logic and truth (values) -- 2.1. Propositions -- 2.2. Logical operations and truth tables -- 2.3. Implication -- 2.4. Tautologies and contradictions -- 2.5. Laws of the algebra of propositions -- 2.6. Additional examples -- 2.7. Problems
3. Quantified predicates, rules of inference, and arguments -- 3.1. Predicates and quantifiers -- 3.2. Arguments -- 3.3. Rules of inference -- 3.4. Additional examples -- 3.5. Problems
4. Mathematical proofs -- 4.1. Some background -- 4.2. Proof by contradiction. -- 4.3. Direct proof -- 4.4. Proof by contraposition -- 4.5. Proof conventions -- 4.6. Counterexamples and disproving a claim -- 4.7. Correct proofs and the human element -- 4.8. Additional examples -- 4.9. Problems
5. Intuitive set theory -- 5.1. Set theory -- 5.2. Venn diagrams -- 5.3. Other operations on sets -- 5.4. Basic results -- 5.5. Additional examples -- 5.6. Problems
6. Mathematical induction -- 6.1. Recurrence relations -- 6.2. Additional examples -- 6.3. Problems
7. Functions. -- 7.1. Mathematical functions -- 7.2. Permutations -- 7.3. Additional examples -- 7.4. Problems
8. The integers and beyond -- 8.1. Division on the integers -- 8.2. Prime numbers and prime factorization -- 8.3. Additional examples -- 8.4. Problems
9. Counting things -- 9.1. Combinatorial proofs and binomial coefficients -- 9.2. Poker hands and multinomial coefficients -- 9.3. The inclusion-exclusion principle -- 9.4. Counting surjections -- 9.5. Additional examples -- 9.6. Problems
10. Relations -- 10.1. Equivalence relations and partitions -- 10.2. The integers modulo n -- 10.3. Partial orders -- 10.4. Additional examples -- 10.5. Problems
11. Number bases, number systems, and operations -- 11.1. Number bases -- 11.2. Operations and number systems -- 11.3. Additional examples -- 11.4. Problems
12. Many infinities : cardinal numbers -- 12.1. Cardinality -- 12.2. Cardinal numbers -- 12.3. Transfinite cardinals -- 12.4. Cardinal results -- 12.5. The Cantor set -- 12.6. Additional examples -- 12.7. Problems
13. Many infinities : ordinal numbers -- 13.1. Partially ordered sets revisited -- 13.2. Partially ordered sets expanded -- 13.3. Limiting elements -- 13.4. Enumeration and isomorphism for ordered sets -- 13.5. Order types of totally ordered sets -- 13.6. Well-ordered sets -- 13.7. Ordinal numbers -- 13.8. Transfinite induction -- 13.9. The well-ordering theorem and the axiom of choice -- 13.10. Additional examples -- 13.11. Problems
14. Paradoxes and axiomatic set theory -- 14.1. Paradoxes of naive set theory -- 14.2. Cantor's paradox -- 14.3. Russel's paradox -- 14.4. An unstated assumption -- 14.5. Axiomatic set theory -- 14.6. Axiomatic theories -- 14.7. Zermelo-Fraenkel set theory and the axiom of choice -- 14.8. Remarks on axiomatic set theory vs. Naive set theory -- 14.9. Gödel's incompleteness theorems -- 14.10. Additional examples -- 14.11. Problems.
Abstract: This text is intended as an introduction to mathematical proofs for students. It is distilled from the lecture notes for a course focused on set theory subject matter as a means of teaching proofs. Chapter 1 contains an introduction and provides a brief summary of some background material students may be unfamiliar with. Chapters 2 and 3 introduce the basics of logic for students not yet familiar with these topics. Included is material on Boolean logic, propositions and predicates, logical operations, truth tables, tautologies and contradictions, rules of inference and logical arguments. Chapter 4 introduces mathematical proofs, including proof conventions, direct proofs, proof-by-contradiction, and proof-by-contraposition. Chapter 5 introduces the basics of naive set theory, including Venn diagrams and operations on sets. Chapter 6 introduces mathematical induction and recurrence relations. Chapter 7 introduces set-theoretic functions and covers injective, surjective, and bijective functions, as well as permutations. Chapter 8 covers the fundamental properties of the integers including primes, unique factorization, and Euclid's algorithm. Chapter 9 is an introduction to combinatorics; topics included are combinatorial proofs, binomial and multinomial coefficients, the Inclusion-Exclusion principle, and counting the number of surjective functions between finite sets. Chapter 10 introduces relations and covers equivalence relations and partial orders. Chapter 11 covers number bases, number systems, and operations. Chapter 12 covers cardinality, including basic results on countable and uncountable infinities, and introduces cardinal numbers. Chapter 13 expands on partial orders and introduces ordinal numbers. Chapter 14 examines the paradoxes of naive set theory and introduces and discusses axiomatic set theory. This chapter also includes Cantor's Paradox, Russel's Paradox, a discussion of axiomatic theories, an exposition on Zermelo-Fraenkel Set Theory with the Axiom of Choice, and a brief explanation of Gödel's Incompleteness Theorems.
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Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader.

Part of: Synthesis digital library of engineering and computer science.

Includes bibliographical references (page 221) and index.

1. Introduction and review of background material -- 1.1. Polynomials and series -- 1.2. Representing numbers -- 1.3. Simple counting -- 1.4. Problems

2. Boolean logic and truth (values) -- 2.1. Propositions -- 2.2. Logical operations and truth tables -- 2.3. Implication -- 2.4. Tautologies and contradictions -- 2.5. Laws of the algebra of propositions -- 2.6. Additional examples -- 2.7. Problems

3. Quantified predicates, rules of inference, and arguments -- 3.1. Predicates and quantifiers -- 3.2. Arguments -- 3.3. Rules of inference -- 3.4. Additional examples -- 3.5. Problems

4. Mathematical proofs -- 4.1. Some background -- 4.2. Proof by contradiction. -- 4.3. Direct proof -- 4.4. Proof by contraposition -- 4.5. Proof conventions -- 4.6. Counterexamples and disproving a claim -- 4.7. Correct proofs and the human element -- 4.8. Additional examples -- 4.9. Problems

5. Intuitive set theory -- 5.1. Set theory -- 5.2. Venn diagrams -- 5.3. Other operations on sets -- 5.4. Basic results -- 5.5. Additional examples -- 5.6. Problems

6. Mathematical induction -- 6.1. Recurrence relations -- 6.2. Additional examples -- 6.3. Problems

7. Functions. -- 7.1. Mathematical functions -- 7.2. Permutations -- 7.3. Additional examples -- 7.4. Problems

8. The integers and beyond -- 8.1. Division on the integers -- 8.2. Prime numbers and prime factorization -- 8.3. Additional examples -- 8.4. Problems

9. Counting things -- 9.1. Combinatorial proofs and binomial coefficients -- 9.2. Poker hands and multinomial coefficients -- 9.3. The inclusion-exclusion principle -- 9.4. Counting surjections -- 9.5. Additional examples -- 9.6. Problems

10. Relations -- 10.1. Equivalence relations and partitions -- 10.2. The integers modulo n -- 10.3. Partial orders -- 10.4. Additional examples -- 10.5. Problems

11. Number bases, number systems, and operations -- 11.1. Number bases -- 11.2. Operations and number systems -- 11.3. Additional examples -- 11.4. Problems

12. Many infinities : cardinal numbers -- 12.1. Cardinality -- 12.2. Cardinal numbers -- 12.3. Transfinite cardinals -- 12.4. Cardinal results -- 12.5. The Cantor set -- 12.6. Additional examples -- 12.7. Problems

13. Many infinities : ordinal numbers -- 13.1. Partially ordered sets revisited -- 13.2. Partially ordered sets expanded -- 13.3. Limiting elements -- 13.4. Enumeration and isomorphism for ordered sets -- 13.5. Order types of totally ordered sets -- 13.6. Well-ordered sets -- 13.7. Ordinal numbers -- 13.8. Transfinite induction -- 13.9. The well-ordering theorem and the axiom of choice -- 13.10. Additional examples -- 13.11. Problems

14. Paradoxes and axiomatic set theory -- 14.1. Paradoxes of naive set theory -- 14.2. Cantor's paradox -- 14.3. Russel's paradox -- 14.4. An unstated assumption -- 14.5. Axiomatic set theory -- 14.6. Axiomatic theories -- 14.7. Zermelo-Fraenkel set theory and the axiom of choice -- 14.8. Remarks on axiomatic set theory vs. Naive set theory -- 14.9. Gödel's incompleteness theorems -- 14.10. Additional examples -- 14.11. Problems.

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This text is intended as an introduction to mathematical proofs for students. It is distilled from the lecture notes for a course focused on set theory subject matter as a means of teaching proofs. Chapter 1 contains an introduction and provides a brief summary of some background material students may be unfamiliar with. Chapters 2 and 3 introduce the basics of logic for students not yet familiar with these topics. Included is material on Boolean logic, propositions and predicates, logical operations, truth tables, tautologies and contradictions, rules of inference and logical arguments. Chapter 4 introduces mathematical proofs, including proof conventions, direct proofs, proof-by-contradiction, and proof-by-contraposition. Chapter 5 introduces the basics of naive set theory, including Venn diagrams and operations on sets. Chapter 6 introduces mathematical induction and recurrence relations. Chapter 7 introduces set-theoretic functions and covers injective, surjective, and bijective functions, as well as permutations. Chapter 8 covers the fundamental properties of the integers including primes, unique factorization, and Euclid's algorithm. Chapter 9 is an introduction to combinatorics; topics included are combinatorial proofs, binomial and multinomial coefficients, the Inclusion-Exclusion principle, and counting the number of surjective functions between finite sets. Chapter 10 introduces relations and covers equivalence relations and partial orders. Chapter 11 covers number bases, number systems, and operations. Chapter 12 covers cardinality, including basic results on countable and uncountable infinities, and introduces cardinal numbers. Chapter 13 expands on partial orders and introduces ordinal numbers. Chapter 14 examines the paradoxes of naive set theory and introduces and discusses axiomatic set theory. This chapter also includes Cantor's Paradox, Russel's Paradox, a discussion of axiomatic theories, an exposition on Zermelo-Fraenkel Set Theory with the Axiom of Choice, and a brief explanation of Gödel's Incompleteness Theorems.

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