Video Lectures
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Chapter 1. Logic and Proofs
Section 1.1  Propositions and Connectives
part 1
part 2  Connectives, Propositional Forms, and Truth Tables.
part 3  Negation and Useful Denials.
Section 1.2  Conditonals and Biconditionals
part 1  Definition of a conditional.
part 2
Useful denial of a conditional.
part 3
Contrapositive, converse, and biconditional.
Section 1.3  Quantifiers
part 1   Open sentences and their truth sets.
part 2
Examples of the use of quantifiers.
part 3
Useful denials of: for all x in U, P(x); there exists x in U such that P(x).
part 4
A few exercises from 1.3.
Section 1.4  Basic Proof Methods I
part 1   Structure of a direct proof of:  for all x in U, P(x) implies Q(x).
part 2
First working definitions.
part 3
First example of a direct proof.
part 4
Second example of a direct proof.
part 5   A direct proof involving inequalities.
part 6   A direct proof involving cases.
part 7   Another direct proof involving cases.

Section 1.5  Basic Proof Methods II

part 1   Structure of  proofs by contraposition and contradiction.
part 2   Example of a proof by contraposition.
part 3   Second example of a proof by contraposition.
part 4   Third example of a proof by contraposition.
part 5   Fourth example of a proof by contraposition.
part 6   First proof by contradiction.
part 7   Second proof by contradiction.
part 8   Alternate ways to express conditionals; biconditional.
part 9   Primes and The Fundamental Theorem of Arithmetic.
part 10   Example of a proof of a biconditional proposition.

Section 1.6  More proofs involving quantifiers
part 1   Constructive and nonconstructive proofs of ``there exists x such that P(x)''.
part 2   Fundamental Theorem of Algebra, Intermediate Value Theorem, and an application to a nonconstructive existence proof.
part 4   Example involving the limit of a function.
part 5   Working definition of limit of a function; the Triangle Inequality and the Reverse Triangle Inequality.
part 6   Exercises involving the Triangle Inequality and the Reverse Triangle Inequality.
part 7   Second example involving the limit of a function.
part 8   Third example involving the limit of a function.
part 9   Fourth example involving the limit of a function.
part 10   Working definition of limit of an infinite sequence and an example.
part 11   A proof that a given sequence does not converge to a given real number.
part 12   Working definition that an infinite sequence diverges to infinity.
part 13   How should you respond to a statement written in symbols?  It depends on whether you wish to merely read it, or to write a proof of it.
part 14   Completion of the exercise started in the previous video.
part 15   Limit proof where you need to calculate both an upper bound and a positive lower bound.
Chapter 2. Set Theory

Section 2.1  Basic Concepts of Set Theory

part 1    Basic set theory.
part 2    Subset of a set.
part 3    The power set of a set.
part 4    A few examples of proofs with sets.

Section 2.2  Set Operations

part 1    Basic set operations.
part 2    Example of a set-theoretic proof.
part 3    Using Venn diagrams to provide examples of various set relationships.
part 4    Use of a Venn diagram to suggest a counterexample.
part 5    Proof of a distributive law in set theory.
part 6    Cartesian product of sets.
part 7    Proofs involving Cartesian product of sets.
part 8   More proofs involving Cartesian products of sets.

Section 2.3  Extended Set Operations and Indexed Families of Sets

part 1    Intersection and union of indexed families of sets.
part 2    Distributive laws for indexed families of sets.
part 3    De Morgan laws for indexed families of sets.
part 4    A simple but useful result.
part 5    A proof involving a denumerable union of intervals.
part 6    A proof involving a denumerable intersection of intervals.

Section 2.4  The Principle of Mathematical Induction (PMI)

part 1    Statement of the PMI and formats of proofs involving PMI.
part 2    Use of the PMI to define functions on the set of natural numbers.
part 3    First example of a proof using  the PMI.
part 4    Second example of a proof using  the PMI.
part 5    Third example of a proof using  the PMI.
part 6    Fourth example of a proof using  the PMI (use of Pascal's triangle).
part 7    Generalized PMI.
part 8    An incorrect proof using PMI.

Section 2.5  Other Forms of Induction

part 1    Two Additional Properties of the set of natural numbers - The Principle of Complete Induction (PCI) and the Well-Ordering Property of the natural numbers..
part 2    An incorrect proof using PCI.
part 3    First example of a  proof using PCI - The Fibonacci numbers.
part 4    Second example of a proof using PCI.
part 5    Third example of a proof using PCI.
part 6    An application of PCI - Proof of the existence part of the Fundamental Theorem of Arithmetic.
part 7    Proof of the Division Algorithm using PCI.
part 8    An application of the Well-Ordering Property - proof that the square root of 2 is irrational.
part 9    An application of the Well-Ordering Property and the Division Algorithm - greatest common divisors.
part 10    More on gcd's; relatively prime integers; Euclid's Lemma.
part 11    Another application of the Division Algorithm - modular arithmetic.
part 12    Logical equivalence of PMI, PCI, and the Well-Ordering Property of the natural numbers.
part 13   Use of PMI to prove that every nonempty, upper bounded subset of the set of natural numbers has a biggest element.
Chapter 3. Relations and Partitions

Section 3.1  Cartesian Products and Relations

part 1    Definition of relation; the identity relation.
part 2
Inverse of a relation.
part 3    Composition of relations.
part 4
Some general theorems involving relations.

Section 3.2  Equivalence Relations
part 1    Definition of equivalence relation; a few examples.
part 2
Example of an equivalence relation.
part 3
Equivalence classes of an equivalence relation.
part 4
Example of equivalence classes - construction of the integers from the natural numbers (intro).
part 5    The equivalence classes of modular arithmetic.

Section 3.3  Partitions and the Equivalence Class Theorem

part 1
Definition of partition of a set and some examples.
part 2
The Equivalence Class Theorem: statement and proof.
part 3
Example of the Equivalence Class Theorem.
part 4    Application of the Equivalence Class Theorem:  Completion of the construction of the integers from the natural numbers.

Section 3.4  Ordering Relations

part 1
Definition of partial orders and linear orders, and a few examples.
part 2    Definition of least upper bound (supremum) and greatest lower bound (infimum), and a few examples.
part 3    Statement of the Completeness Property of the set of real numbers.
part 4    An application of the Completeness Property of the set of real numbers: proof of the existence of the square root of 2.
Chapter 4. Functions

Section 4.1  Functions as Relations

part 1    Functions: Basic definitions.
part 2    Some special types of functions.

Section 4.2  Construction of Functions
part 1    Inverse of functions.
part 2    Composition of functions, and the associative law of composition for functions.
part 3    Inverse functions and composition.
part 4    Restriction and extension of functions.

Section 4.3 and 4.4  Onto Functions; One-to-One Functions; One-to-One Correspondences and Inverse Functions
part 1    Definition of  onto (surjective) functions and some examples.
part 2    An example involving surjectivity.
part 3    Another example involving surjectivity.
part 4    Yet another example involving surjectivity.
part 5    Definition of one-to-one (injective) functions and some examples.
part 6    An example involving injectivity.
part 7    Bijections: working definition of bijection and a theorem involving injections, surjections, and composition of functions.
part 8    Bijections:  A theorem about bijections and inverse.

Section 4.5  Images and Inverse Images of Sets
part 1    Working definitions of image and inverse image of sets, and a simple example..
part 2    An example involving the image of a set.
part 3    An example involving the inverse image of a set..
part 4    A general theorem concerning inverse images, unions of sets, and intersections of sets.
part 5    A general theorem concerning images, unions of sets, and intersections of sets.

Chapter 5.  Cardinality

Section 5.1(a)  Equivalent Sets
(This is the first series of lectures on Section 5.1)
part 1
Fundamental definition that two sets have the same cardinality.
part 2    A few examples involving cardinality of sets.
part 3    Another example involving cardinality of sets.
part 4    Cardinality as an equivalence relation.
part 5
Some general results involving cardinality (valid for all sets).

Section 5.1(b)  Finite Sets
(This is the second series of lectures on Section 5.1)
part 1
Definition of finite sets and some theorems.
part 2    Theorems converning finite sets: unions, subsets, differences, Cartesian products.

Section 5.2-5.5  Infinite Sets
part 1
Definition of  infinite, denumerable, countable, and uncountable  sets, and a few simple examples of denumerable sets.
part 2
A surprising result- The Cartesian product of denumerable sets is denumerable.
part 3
Proof that the set of integers is denumerable.
part 4
Proof that any infinite subset of  the natural numbers is denumerable.
part 5
Any infinite subset of a denumerable set is denumerable.
part 6
The denumerable union of denumerable sets is denumerable.
part 7
The set of rational numbers is denumerable.
part 8
Cantor's diagonlization argument that (0,1) is uncountable.
part 9
The Cantor-Schroder-Bernstein Theorem and a few examples.
part 10   R x R and R have the same cardinality.
part 11   The comparison of cardinalities: what does it mean to say a set has strictly greater cardinality than another set?
part 12   Cantor's Theorem: Given any nonempty set, there exists a set of strictly bigger cardinality.
part 13   Proof of the Cantor-Schroder-Bernstein Theorem.
part 14
Algebraic and transcendental numbers.
part 15
The Axiom of Choice and Zorn's Lemma.
part 16
Proof of the Comparability Theorem :  Application of Zorn's Lemma.

Section 5.1(c)  Proofs of Theorems Concerning Finite sets. With these lectures I fill in the proofs of results concerning finite sets stated in Section 5.1(b).
part 1
Proof of the Pigeonhole Principle.
part 2   Definition of Finite Set: The main result of this lecture together with the Pigeonhole Principle allow us to define precisely what it means to say that a set is finite.
part 3
Cardinality of the disjoint union of finitely many finite sets. The results in this lecture play a key role in the remaining lectures.
part 4
Cardinality of subsets of finite sets.
part 5
Cardinality of the difference of finite sets.
part 6   Cardinality of the union of two finite sets.
part 7   Cardinality of the Cartesian product of two finite sets.