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Subject:	Mathematics
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Math Doctor Bob

Name:	Math Doctor Bob (Robert Donley)
Type:	Individual

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Math Major Basics: Prerequisites to Higher Mathematics

Video Lectures

Displaying all 17 video lectures.

Lecture 1 Play Video	Propositional Logic Basic Methods: We introduce tools need for the construction of proofs. In this part, we consider operation for creating new mathematical statements from old ones. We also introduce truth tables.
Lecture 2 Play Video	Logical Equivalence Basic Methods: We define tautology and contradiction and consider the conditions of logical equivalence and implication. Examples include DeMorgan's Laws for logic, modus ponens, and the Law of the Excluded Middle. As a final note, we introduce the Substitution Rules.
Lecture 3 Play Video	Formal Proofs Basic Methods: We define theorems and describe how to formally construct a proof. We note further rules of inference and show how the logical equivalence of reductio ad absurdum allows proof by contradiction.
Lecture 4 Play Video	Methods of Proof Basic Methods: We note the different methods of informal proof, which include direct proof, proof by contradiction, and proof by induction. We give proofs that sqrt(2) is irrational and that there are infinitely many primes, among others.
Lecture 5 Play Video	Example of Complete/Strong Induction Basic Methods: As an example of complete induction, we prove the Binet formula for the Fibonacci numbers.
Lecture 6 Play Video	Example of Induction (Advanced): Balanced Strings Math Major Basics: Using complete induction, we show that, for balanced strings, the number of left parentheses equals the number of right parentheses. A balanced string is a finite sequence of (s and )s constructed recursively.
Lecture 7 Play Video	Naive Set Theory Basic Methods: We introduce basic notions from naive set theory, including sets, elements, and subsets. We give examples of showing two sets are equal by mutual inclusion. Then we define the power set and note Russell's paradox.
Lecture 8 Play Video	Set Operations Basic Methods: We introduce the basic set operations of union, intersection, and complement, which mirror the logical constructions of or, and, and not. We note the main laws for these set operations and give more examples of double inclusion proofs. Finally we consider indexed families of sets and logical quantifiers.
Lecture 9 Play Video	Two Tough Set Equalities Basic Methods: We give two examples of set equality proofs using union and intersections. Given a family of subsets A_k in X, we consider set constructions that indicate the set of all elements in X that are in infinitely many A_k and the set of elements that are in all but finitely many A_k.
Lecture 10 Play Video	Binary Relations Basic Methods: We define the Cartesian product of two sets X and Y and use this to define binary relations on X. We explain the properties of reflexive, symmetric, transitive, anti-symmetric, and anti-reflexive. In turn, these lead to partially ordered set and equivalence relations.
Lecture 11 Play Video	Mappings 1: Examples Basic Methods: We define mappings (or functions) between sets and consider various examples. These include binary operations, projections, and quotient maps. We show how to construct the rational numbers from the integers and explain why division by zero is a forbidden operation.
Lecture 12 Play Video	Mappings 2: Basic Properties Basic Methods: We continue with properties of mappings. We define domain, range, image, and inverse image. Some rules for set operations and noted, and then the notions of one-one and onto are introduced. Examples are given, and finally we further identify equivalence relations and partitions with onto mappings.
Lecture 13 Play Video	Mappings 3: Composition and Inverse Mappings Basic Methods: We define composition of mappings and draw parallels to multiplication of real numbers. Items include associativity, identity, and commutativity. Consideration of multiplicative inverses leads to the definition of an inverse mapping, and we give conditions for its existence.
Lecture 14 Play Video	Cardinality 1: Finite Sets Basic Methods: We define cardinality as an equivalence relation on sets using one-one correspondences. In this talk, we consider finite sets and counting rules.
Lecture 15 Play Video	Cardinality 2: Infinite Sets Basic Methods: We continue the study of cardinality with infinite sets. First the class of countably infinite sets is considered, and basic results given. Then we give examples of uncountable sets using Cantor diagonalization arguments.
Lecture 16 Play Video	Divisibility Properties of the Integers Basic Methods: We develop basic properties of the integers, with a focus on divisibility. Main results include Bezout's identity, unique factorization of integers into primes, and the definition of modular integers.
Lecture 17 Play Video	The Euclidean Algorithm for the Integers Basic Methods: Revisiting the proof of Bezout's Identity, we give an algorithm for computing gcd(m, n) without factoring m and n. In turn, the Euclidean algorithm provides a method for finding the coefficients in Bezout's identity.