Announcements

Lecture No	Date	Content	Corresponding Reading material from textbook
1	Sep. 18	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	Sep. 20	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	Sep. 25	Compound statements, contradictions and tautologies, propositional calculus.	Pages 10-20
4	Sep. 27	Theorem types and proof methods: Existence, uniqueness, and classification theorems; Proving Implication: trivial, direct, contrapositive, and deductive proofs; proof by cases	Pages 23-27, 30
5	Oct. 02	Proof by contradiction; proof by induction; Induction Axiom and Principle of Mathematical Induction; an application	Pages 27-29, 31-33
6	Oct. 04	Another example of proof by induction; Complete Induction; Recursive definitions; Characterization theorems	Pages 35-41
7	Oct. 09	Sets, axioms of extensionality, existence, and specification; empty set, subsets, power set axiom, Russell's paradox	Pages 45-49
8	Oct. 16	Intersection of two sets, disjoint sets, universal sets, 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 and its properties	Pages 49-55
9	Oct. 18	Union axiom and properties of the union of a collection of sets, Axiom of pairing, ordered pairs, Cartesian product of two and a finite list of sets	Pages 55-60
10	Oct. 23	Successor of a set, construction of natural numbers, inductive sets, definition of the set of natural numbers, the addition and multiplication of natural numbers and their ordering; Definition and examples of a relation	Pages 60-62 & 67
11	Oct. 25	Image and inverse image of sets under a relation, domain and range of a relation, equal relations, image and inverse image of the intersection and union of collections of subsets under a relation, reflexive, symmetric, antisymmetric, and transitive relations	Pages 68-73
12	Oct. 30	Restriction and extension of relations, identity relation, inverse relation; Composition of relations, domain of a composite relation, associativity and non-commutativity of composition of relations, inverse of a composite relation	Pages 73-78
13	Nov. 01	Equivalence relations, congruence relation, 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	Pages 78-82
14	Nov. 06	Partial and total ordering relations, inverse of a partial ordering relation, posets, graphical representation of posets, examples	Pages 82-88
15	Nov. 08	Upper and lower bounds, supremum and infimum, chains and Zorn's Lemma, total ordering of natural numbers (without proof), well-ordering axiom (without proof), statement of the well-ordering principle (theorem), an application of well-ordering axiom: "gcd(m,n) may be expressed as a linear combination of m and n with integer coefficients," and a corollary of this theorem	Pages 88-92
16	Nov. 13	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, one-to-one and onto functions, bijections, inclusion maps	Pages 97-102
17	Nov. 15	Images and inverse images of intersection and complement of sets under a general and a one-to-one function, Composition of two functions and their domain, composition of one-to-one and onto functions	Pages 103-108
18	Nov. 20	Invertible functions and their characterization, inverse of the inverse function, and inverse of the composition of two invertible functions	Pages 108-110
19	Nov. 22	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; Characterization of one-to-one, onto, and bijective functions mapping I_m to I_n	Pages 110-114
20	Nov. 29	Sequences: Definition and examples, sequences with distinct terms, increasing and decreasing sequences, subsequences of a sequence; Real sequences and their convergence, real series, examples	Pages 114-118
21	Nov. 30	Some basic results on real series, geometric series and its limit, binary expansion of real numbers; Equivalent sets and their properties, N~Z	Pages 119-120, 125-126
22	Dec. 04	Finite sets, order of a finite set, subsets of a finite set; 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 126-132
23	Dec. 06	Infinite sets and their characterizations, subsets and cross product of infinite sets, countably infinite sets and their characterization, examples	Pages 132-135
24	Dec. 11	Subsets, union, and Cartesian product of countable sets; Countability of rational numbers Q, and Q^n for all positive integers n	Pages 136-139
25	Dec. 13	1-to-1 mappings between finite, countably infinite and infinite sets; Uncountable sets: Thm: R is uncountable, unions and Cartesian product of uncountable sets; Cardinal numbers: 1-to-1 correspondence between cardinal numbers of finite sets and the natural numbers; Motivation for ordering cardinal numbers	Pages 139-143
26	Dec. 18	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)); Cantor's paradox	Pages 143-147
27	Dec. 25	Cardinal arithmetic: Addition and multiplication of cardinal numbers, Derivation of the cardinality of R, R^2, and R^n; Continuum and Generalized Continuum hypotheses	Pages 147-151
28	Dec. 27	Cartesian product of an arbitrary family of sets, Axiom of choice, Thm: If f:A->B is onto, Card(A) ³ Card(B)	Pages 151-153

Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.