An Introduction to Mathematical Reasoning by Eccles, Peter J. -- Read -- Imperial Library of Trantor

Index

Cover Title Copyright Epigraph Dedication Contents Preface Part I: Mathematical statements and proofs

1 The language of mathematics

1.1 Mathematical statements 1.2 Logical connectives Exercises

2 Implications

2.1 Implications 2.2 Arithmetic 2.3 Mathematical truth Exercises

3 Proofs

3.1 Direct proofs 3.2 Constructing proofs backwards Exercises

4 Proof by contradiction

4.1 Proving negative statements by contradiction 4.2 Proving implications by contradiction 4.3 Proof by contrapositive 4.4 Proving ‘or’ statements Exercises

5 The induction principle

5.1 Proof by induction 5.2 Changing the base case 5.3 Definition by induction 5.4 The strong induction principle Exercises

Problems I

Part II: Sets and functions

6 The language of set theory

6.1 Sets 6.2 Operations on sets 6.3 The power set Exercises

7 Quantifiers

7.1 Universal statements 7.2 Existential statements 7.3 Proving statements involving quantifiers 7.4 Disproving statements involving quantifiers 7.5 Proof by induction 7.6 Predicates involving more that one free variable 7.7 The Cartesian product of two sets Exercises

8 Functions

8.1 Functions and formulae 8.2 Composition of functions 8.3 Sequences 8.4 The image of a function 8.5 The graph of a function Exercises

9 Injections, surjections and bijections

9.1 Properties of functions 9.2 Bijections and inverses 9.3 Functions and subsets 9.4 Peano’s axioms for the natural numbers Exercises

Problems II

Part III: Numbers and counting

10 Counting

10.1 Counting finite sets 10.2 Two basic counting principles 10.3 The inclusion-exclusion principle Exercises

11 Properties of finite sets

11.1 The pigeonhole principle 11.2 Finite sets of real numbers 11.3 Two applications of finiteness The greatest common divisor Exercises

12 Counting functions and subsets

12.1 Counting sets of functions 12.2 Counting sets of subsets 12.3 The binomial theorem Exercises

13 Number systems

13.1 The rational numbers 13.2 The irrationality of 13.3 Real numbers and infinite decimals Exercises

14 Counting infinite sets

14.1 Countable sets 14.2 Denumerable sets 14.3 Uncountable sets Exercises

Problems III

Part IV: Arithmetic

15 The division theorem

15.1 The division theorem 15.2 Some applications Exercises

16 The Euclidean algorithm

16.1 Finding the greatest common divisor 16.2 The Euclidean algorithm Exercises

17 Consequences of the Euclidean algorithm

17.1 Integral linear combinations 17.2 An alternative definition of the greatest common divisor 17.3 Coprime pairs Exercises

18 Linear diophantine equations

18.1 Diophantine equations 18.2 A condition for the existence of solutions 18.3 Finding all the solutions − the homogeneous case 18.4 Finding all the solutions – the general case Exercises

Problems IV

Part V: Modular arithmetic

19 Congruence of integers

19.1 Basic definitions 19.2 The remainder map 19.3 Division in congruences Exercises

20 Linear congruences

20.1 A criterion for the existence of solutions 20.2 Linear congruences and diophantine equations Exercises

21 Congruence classes and the arithmetic of remainders

21.1 Congruence classes 21.2 The arithmetic of congruence classes 21.3 The arithmetic of remainders 21.4 Linear diophantine equations Exercises

22 Partitions and equivalence relations

22.1 Partitions 22.2 Equivalence relations 22.3 Equivalence relations and partitions Exercises

Problems V

Part VI: Prime numbers

23 The sequence of prime numbers

23.1 Definition and basic properties 23.2 The sieve of Eratosthenes 23.3 The fundamental theorem of arithmetic 23.4 Applications of the fundamental theorem of arithmetic 23.5 The distribution of prime numbers Exercises

24 Congruence modulo a prime

24.1 Fermat’s little theorem 24.2 Wilson’s theorem 24.3 Looking for primes Exercises

Problems VI

Solutions to exercises Bibliography List of symbols Index

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