Section 1.1 Propositions and Connectives

part 1

part 2Connectives, Propositional Forms, and Truth Tables.

part 3Negation and Useful Denials.

Section 1.2 Conditonals and Biconditionals

part 1Definition of a conditional.Useful denial of a conditional.

part 2Contrapositive, converse, and biconditional.

part 3

Section 1.3 Quantifiers

part 1Open sentences and their truth sets.Examples of the use of quantifiers.

part 2Useful denials of: for all x in U, P(x); there exists x in U such that P(x).

part 3A few exercises from 1.3.

part 4

Section 1.4 Basic Proof Methods I

part 1Structure of a direct proof of: for all x in U, P(x) implies Q(x).First working definitions.

part 2First example of a direct proof.

part 3Second example of a direct proof.

part 4

part 5A direct proof involving inequalities.

part 6A direct proof involving cases.

part 7Another direct proof involving cases.

Section 1.5 Basic Proof Methods II

part 1Structure of proofs by contraposition and contradiction.

part 2Example of a proof by contraposition.

part 3Second example of a proof by contraposition.

part 4Third example of a proof by contraposition.

part 5Fourth example of a proof by contraposition.

part 6First proof by contradiction.

part 7Second proof by contradiction.

part 8Alternate ways to express conditionals; biconditional.

part 9Primes and The Fundamental Theorem of Arithmetic.

part 10Example of a proof of a biconditional proposition.

Section 1.6 More proofs involving quantifiers

part 1Constructive and nonconstructive proofs of ``there exists x such that P(x)''.

part 2Fundamental Theorem of Algebra, Intermediate Value Theorem, and an application to a nonconstructive existence proof.

part 3General comments about existentially quantified statements.

part 4Example involving the limit of a function.

part 5Working definition of limit of a function; the Triangle Inequality and the Reverse Triangle Inequality.

part 6Exercises involving the Triangle Inequality and the Reverse Triangle Inequality.

part 7Second example involving the limit of a function.

part 8Third example involving the limit of a function.

part 9Fourth example involving the limit of a function.

part 10Working definition of limit of an infinite sequence and an example.

part 11A proof that a given sequence does not converge to a given real number.

part 12Working definition that an infinite sequence diverges to infinity.

part 13How 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 14Completion of the exercise started in the previous video.

part 15Limit proof where you need to calculate both an upper bound and a positive lower bound.

Section 2.1 Basic Concepts of Set Theory

part 1Basic set theory.

part 2Subset of a set.

part 3The power set of a set.

part 4A few examples of proofs with sets.

Section 2.2 Set Operations

part 1Basic set operations.

part 2Example of a set-theoretic proof.

part 3Using Venn diagrams to provide examples of various set relationships.

part 4Use of a Venn diagram to suggest a counterexample.

part 5Proof of a distributive law in set theory.

part 6Cartesian product of sets.

part 7Proofs involving Cartesian product of sets.

part 8More proofs involving Cartesian products of sets.

Section 2.3 Extended Set Operations and Indexed Families of Sets

part 1Intersection and union of indexed families of sets.

part 2Distributive laws for indexed families of sets.

part 3De Morgan laws for indexed families of sets.

part 4A simple but useful result.

part 5A proof involving a denumerable union of intervals.

part 6A proof involving a denumerable intersection of intervals.

Section 2.4 The Principle of Mathematical Induction (PMI)

part 1Statement of the PMI and formats of proofs involving PMI.

part 2Use of the PMI to define functions on the set of natural numbers.

part 3First example of a proof using the PMI.

part 4Second example of a proof using the PMI.

part 5Third example of a proof using the PMI.

part 6Fourth example of a proof using the PMI (use of Pascal's triangle).

part 7Generalized PMI.

part 8An incorrect proof using PMI.

Section 2.5 Other Forms of Induction

part 1Two Additional Properties of the set of natural numbers - The Principle of Complete Induction (PCI) and the Well-Ordering Property of the natural numbers..

part 2An incorrect proof using PCI.

part 3First example of a proof using PCI - The Fibonacci numbers.

part 4Second example of a proof using PCI.

part 5Third example of a proof using PCI.

part 6An application of PCI - Proof of the existence part of the Fundamental Theorem of Arithmetic.

part 7Proof of the Division Algorithm using PCI.

part 8An application of the Well-Ordering Property - proof that the square root of 2 is irrational.

part 9An application of the Well-Ordering Property and the Division Algorithm - greatest common divisors.

part 10More on gcd's; relatively prime integers; Euclid's Lemma.

part 11Another application of the Division Algorithm - modular arithmetic.

part 12Logical equivalence of PMI, PCI, and the Well-Ordering Property of the natural numbers.

part 13Use of PMI to prove that every nonempty, upper bounded subset of the set of natural numbers has a biggest element.

Section 3.1 Cartesian Products and Relations

part 1Definition of relation; the identity relation.Inverse of a relation.

part 2

part 3Composition of relations.Some general theorems involving relations.

part 4

part 1Definition of equivalence relation; a few examples.Example of an equivalence relation.

part 2Equivalence classes of an equivalence relation.

part 3Example of equivalence classes - construction of the integers from the natural numbers (intro).

part 4

part 5The equivalence classes of modular arithmetic.

Section 3.3 Partitions and the Equivalence Class Theorem

Definition of partition of a set and some examples.

part 1The Equivalence Class Theorem: statement and proof.

part 2Example of the Equivalence Class Theorem.

part 3

part 4Application of the Equivalence Class Theorem: Completion of the construction of the integers from the natural numbers.

Section 3.4 Ordering Relations

Definition of partial orders and linear orders, and a few examples.

part 1

part 2Definition of least upper bound (supremum) and greatest lower bound (infimum), and a few examples.

part 3Statement of the Completeness Property of the set of real numbers.

part 4An application of the Completeness Property of the set of real numbers: proof of the existence of the square root of 2.

Section 4.1 Functions as Relations

part 1Functions: Basic definitions.

part 2Some special types of functions.

Section 4.2 Construction of Functions

part 1Inverse of functions.

part 2Composition of functions, and the associative law of composition for functions.

part 3Inverse functions and composition.

part 4Restriction and extension of functions.

Section 4.3 and 4.4 Onto Functions; One-to-One Functions; One-to-One Correspondences and Inverse Functions

part 1Definition of onto (surjective) functions and some examples.

part 2An example involving surjectivity.

part 3Another example involving surjectivity.

part 4Yet another example involving surjectivity.

part 5Definition of one-to-one (injective) functions and some examples.

part 6An example involving injectivity.

part 7Bijections: working definition of bijection and a theorem involving injections, surjections, and composition of functions.

part 8Bijections: A theorem about bijections and inverse.

Section 4.5 Images and Inverse Images of Sets

part 1Working definitions of image and inverse image of sets, and a simple example..

part 2An example involving the image of a set.

part 3An example involving the inverse image of a set..

part 4A general theorem concerninginverse images, unions of sets, and intersections of sets.

part 5A general theorem concerningimages, unions of sets, and intersections of sets.

Section 5.1(a) Equivalent Sets

part 1

part 5

Section 5.1(b) Finite Sets

part 1

part 1

part 2

part 3

part 4

part 5

part 6

part 7

part 8

part 9

part 14

part 15

part 16

part 1

part 3

part 4

part 5