  # Math Major Basics: Prerequisites to Higher Mathematics

## Video Lectures

Displaying all 17 video lectures.
 Lecture 1 Play Video Propositional LogicBasic 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 EquivalenceBasic 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 ProofsBasic 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 ProofBasic 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 InductionBasic 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 StringsMath 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 TheoryBasic 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 OperationsBasic 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 EqualitiesBasic 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 RelationsBasic 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: ExamplesBasic 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 PropertiesBasic 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 MappingsBasic 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 SetsBasic 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 SetsBasic 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 IntegersBasic 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 IntegersBasic 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.