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PLP:
An introduction to mathematical proof
Seçkin Demirbaş, Andrew Rechnitzer
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\(\require{cancel}\require{upgreek} \newcommand{\ds}{\displaystyle} \renewcommand{\textcolor}[2]{{\color{#1}{#2}}} \newcommand{\es}{ {\varnothing}} \newcommand{\st}{ \;\mathrm{s.t.}\; } \newcommand{\so}{ \;\mid\; } \newcommand{\pow}[1]{ \mathcal{P}\left(#1\right) } \newcommand{\set}[1]{ \left\{#1\right\} } \renewcommand{\neg}{\sim} \newcommand{\rel}[1][R]{\;\mathcal{#1}\;} \newcommand{\nrel}[1][R]{\;\cancel{\mathcal{#1}}\;} \newcommand{\dee}[1]{\mathrm{d}#1} \newcommand{\diff}[2]{\dfrac{\mathrm{d}#1}{\mathrm{d}#2}} \renewcommand{\mod}[1]{\ (\mathrm{mod}\ #1)} \newcommand{\lcm}{\mathrm{lcm}} \let\oldepsilon\epsilon \renewcommand{\epsilon}{\varepsilon} \newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Dedication
Acknowledgements
Preface
1
Sets
1.1
Not so formal definition
1.1.1
Who is this reader you keep on mentioning?
1.2
Describing a set
1.3
Onward
1.4
Exercises
2
A little logic
2.1
Statements and open sentences
2.2
Negation
2.3
Or and And
2.4
The implication
2.5
Modus ponens and chaining implications
2.5.1
Modus ponens
2.5.2
Affirming the consequent and denying the antecedent
2.5.3
Chaining implications together
2.6
The converse, contrapositive and biconditional
2.7
Exercises
3
Direct proofs
3.1
Trivialities and vacuousness
3.2
Direct proofs
3.3
Proofs of inequalities
3.4
A quick visit to disproofs
3.5
Exercises
4
More logic
4.1
Tautologies and contradictions
4.2
Logical equivalence
4.3
Exercises
5
More proofs
5.1
Contrapositive
5.2
Proofs with cases
5.3
Congruence modulo
\(n\)
5.4
Absolute values and the triangle inequality
5.5
Exercises
6
Quantifiers
6.1
Quantified statements
6.2
Negation of quantifiers
6.3
Nested quantifiers
6.4
Quantifiers and rigorous limits
6.4.1
Convergence of sequences
6.4.1.1
Quantifying towards a definition
6.4.1.2
Some examples
6.4.2
The limit of a function
6.5
(Optional) Properties of limits
6.5.1
(Optional) Some properties of limits of sequences
6.5.1.1
Uniqueness of limits
6.5.1.2
Linearity of limits
6.5.1.3
Product of limits
6.5.1.4
Ratio of limits
6.5.2
(Optional) Some properties of limits of functions
6.5.2.1
Uniqueness of limits
6.5.2.2
Linearity of limits
6.5.2.3
Product of limits
6.5.2.4
Ratio of limits
6.6
Exercises
7
Induction
7.1
Induction
7.2
More general inductions
7.2.1
A little more general
7.2.2
More general and yet equivalent
7.3
Exercises
8
Return to sets
8.1
Subsets
8.2
Set operations
8.3
Cartesian products of sets
8.4
Some set-flavoured results
8.5
Indexed sets
8.6
Exercises
9
Relations
9.1
Relations
9.2
Properties of relations
9.3
Equivalence relations & classes
9.4
Congruence revisited
9.5
Greatest divisors, Bézout and the Euclidean algorithm
9.6
Uniqueness of prime factorisation
9.7
Exercises
10
Functions
10.1
Functions
10.2
A more abstract definition
10.3
Images and preimages of sets
10.4
Injective and surjective functions
10.4.1
Injections and surjections
10.4.2
Bijective functions
10.5
Composition of functions
10.6
Inverse functions
10.7
(Optional) The axiom of choice
10.8
Exercises
11
Proof by contradiction
11.1
Structure of a proof by contradiction
11.2
Some examples
11.2.1
The irrationality of
\(\sqrt{2}\)
11.2.2
The infinitude of primes
11.3
Exercises
12
Cardinality
12.1
Finite sets
12.1.1
Equinumerous sets, bijections and pigeons
12.1.2
Comparing with functions
12.1.3
Infinite sets are strange.
12.2
Denumerable sets
12.3
Uncountable sets
12.4
Comparing cardinalities
12.4.1
Extending to the infinite
12.4.2
Cantor’s Theorem and infinite infinities
12.4.3
Congratulations are in order
12.4.4
One more question
12.5
More comparisons of cardinalties
12.5.1
The Cantor-Schröder-Bernstein theorem
12.5.2
Applications
12.5.3
Proof that
\(|\pow{\mathbb{N}}| = |\mathbb{R}|\)
12.6
(Optional) Cantor’s first proof of the uncountability of the reals
12.7
Exercises
Backmatter
A
Hints for Exercises
B
Scratchwork for Exercises
C
Solutions to Exercises
Colophon
Colophon
Colophon
This book was authored in PreTeXt.
