Math 103, Spring 2007

Topics Covered in Each Lecture

 

Lecture No

Date

Content

Corresponding Reading material from the textbook

1

 Feb. 06 Mathematical theories: Definitions, axioms, theorems (lemmas, propositions), conjectures, incompleteness theorem of Gödel; Examples: Euclidean and non-Euclidean geometry, Fermat's theorem; Scientific theories: Basic concepts, postulates, predictions, experimentation, technological advances and the development of science. Pages 1-4

2

 Feb. 08 Mathematical language: 4-color-problem, a crude description of sets, defining symbols, some other basic mathematical symbols; Logic: statements and predicates, their equality and logical equivalence, qualifiers, negation of statements, negation of qualified predicates. Pages 4-10

3

 Feb. 13 Compound statements, contradictions and tautologies, propositional calculus. Pages 10-20
4 Feb. 15 Theorem types and proof methods: Existence, uniqueness, and classification theorems; Proving Implication: trivial, direct, contrapositive, and deductive proofs; Proof by contradiction (general idea) Pages 23-28
5 Feb. 20 Applications of proof by contradiction; proof by induction; Induction Axiom and Principle of Mathematical Induction; applications Pages 28-35
6 Feb. 22 Another example of proof by induction; Complete Induction; Recursive definitions; Characterization theorems; Examples and counterexamples.

Pages 35-41

7 Feb. 27 Sets, Axioms of extensionality, existence, specification, empty set, subsets, power set axiom, Russell's paradox, intersection of two sets

Pages 45-50

8 Mar. 01 Disjoint sets, universal set, complement of a set in a universal set; Intersection of more than two sets, indexing sets, collection of sets labeled by the elements of an indexing set, intersection of a collection of sets, union of a collection of sets and the union axiom.

Pages 51-55

9 Mar. 06 Properties of the union of collections of sets, Axiom of pairing, ordered pairs, Cartesian product of two and a finite list of sets

Pages 56-60

10 Mar. 08 Successor of a set, construction of natural numbers, inductive sets, definition of the set of natural numbers, the addition and multiplication of natural numbers; Relations, the image and inverse image of a set under a relation, the domain and range of a relation Pages 60-67
11 Mar. 13 Examples of relations, properties of image and inverse image of sets under relations, equal relations, restriction and extension of relations, image and inverse image of intersection and union of collections of subsets under a relation, reflexive, symmetric, antisymmetric, and transitive relations, identity relation, inverse relation Pages 67-72
12 Mar. 15 Composition of relations, its domain, associativity and non-commutativity of comoposition of relations, inverse of a composite relation, equivalence relations, examples: congruence relation, parallelism in Euclidean plane Pages 72-77
13 Mar. 20 Partitions of a set, equivalence classes and their properties, use of equivalence relations in classification schemes, the equivalence relation associated with a given partition, characterization of all possible equivalence relation on a given set. Partial and total ordering relations, examples. Pages 77-81
14 Mar. 22 Inverse of a partial ordering relation, posets, graphical representation of posets, maximal, minimal, greatest, and least elements of a poset, upper and lower bounds, supremum and infimum, chains and Zorn Lemma Pages 82-88
15 Mar. 27 Total ordering of natural numbers (without proof), well-ordering axiom (without proof), statement of the well-ordering principle (theorem), Application of well-ordering axiom: "gcd(m,n) may be expressed as a linear combination of m and n with integer coefficients," and two corollaries of this theorem; Functions: Motivation, proving that a relation is well-defined, equality of functions, image and inverse image of a set under a function, domain and range of a function, restriction and extensions of a function. Pages 89-98
16 Mar. 29 One-to-one and onto functions, bijections, inclusion maps, images and inverse images of intersection and complement of sets under a general and a one-to-one function. Pages 98-104
17 Apr. 10 Composition of two functions and their domain, composition of one-to-one and onto functions, Invertible functions and their characterization. Pages 105-108
18 Apr. 12 Inverse of the inverse function and inverse of the composition of two invertible functions, Functions acting in I_n:={1,2,...,n}: transpositions, permutations, a characterization theorem for bijections (with proof): A function acting in I_n is a  bijection if and only if it is a permutation. Pages 108-110
19 Apr. 17 Characterization of one-to-one functions, onto functions and bijections mapping I_m to I_n where I_n:={1,2,...,n}. Sequences: Definition and examples, sequences with distinct terms. Pages 110-113
20 Apr. 19 Increasing and decreasing sequences, subsequences of a sequence, real sequences and their convergence, real series, geometric series and its limit. Pages 113-118
21 Apr. 24 Equivalent sets and their properties, N~Z, finite sets, order of a finite set, subsets of a finite set. Pages 123-127
22 Apr. 26 Intersections, union, and Cartesian product of two finite sets; Characterization of inequality between orders of two finite sets in terms of the existence of certain one-to-one or onto functions among them; Non-equivalence of proper subsets of a finite set with that set. Pages 127-130
23 May 01 Infinite sets and their characterizations, subsets and cross product of infinite sets, countably infinite sets and their characterization, examples. Pages 130-133
24 May 03 Subsets, union, and Cartesian product of countable sets; 1-to-1 mappings between finite, countably infinite and infinite sets; proof of the countability of rational numbers Q, and Q^n for all positive integers n. Pages 133-137
25 May 08 Uncountable sets: Thm: R is uncountable, unions and Cartesian product of uncountable sets, R^n is uncountable; Cardinal numbers: 1-to-1 correspondence between cardinal numbers of finite sets and the natural numbers; Motivation for ordering cardinal numbers Pages 137-142
26 May 09 Ordering cardinal numbers: Cantor-Schreoder-Bernstein Theorem (without proof); n< Card(N) for every natural number, Card(N)<Card (R); Thm (Cantor): Card (A)<Card(P(A)); Cardinal arithmetic: Addition and multiplication of cardinal numbers; Cantor's paradox. Pages 142-145
27 May 15 Derivation of the cardinality of R, R^2, and R^n; Continuum and Generalized Continuum hypotheses. Pages 145-149
28 May 17 Cartesian product of an arbitrary family of sets, Axiom of choice, Thm: If f:A->B is onto, Card(A) ³ Card(B). Pages 149-151

Note: The pages from the textbook listed above may not include all the material covered in the lectures.