E0 201, Fall 2015 | Parimal Parag

Proofs and Measures

Lectures

Lecture-01 Informal Logic
Lecture-02 Strategies for Proofs
Lecture-03 Sets
Lecture-04 Set Operations
Lecture-05 Functions: Images, Compositions, Inverses
Lecture-06 Functions: Injectivity, Surjectivitiy, and Bijectivity
Lecture-07 Relations
Lecture-08 Equivalence Relations
Lecture-09 Properties of Natural Numbers
Lecture-10 Recursion
Lecture-11 Cardinality of Sets
Lecture-12 Topological Spaces
Lecture-13 Basis for a Topology
Lecture-14 Order and Product Topologies
Lecture-15 Subspace Topology
Lecture-16 Closed Sets and Limit Points
Lecture-17 Continuous Functions
Lecture-18 Product Topology
Lecture-19 Connectedness
Lecture-20 Compactness
Lecture-21 Measurable Spaces
Lecture-22 Introduction to Measure Theory
Lecture-23 Basic Properties of Measures
Lecture-24 Examples of Measures
Lecture-25 Integration of Non-negative functions
Lecture-26 Dominated Convergence Theorem

Homework

Homework 1 (Due August 25, 2015) Problems from Bloch 2nd Edition. 1.2.8, 1.2.13, 1.3.1, 1.3.2, 1.4.1, 1.5.8, 1.5.10, 2.49, 2.5.7.
Homework 2 (Due August 25, 2015) Problems from Bloch 2nd Edition. 3.2.7, 3.2.13, 3.3.4, 3.3.12 3.4.6, 4.1.1, 4.1.8, 4.2.13, 4.3.5.
Homework 3 (Due September 15, 2015) Problems from Bloch 2nd Edition. 4.4.1, 4.4.3, 4.4.6, 4.4.8, 4.4.2, 4.4.11, 4.5.2, 4.5.4, 5.1.1, 5.1.4, 5.1.9, 5.3.2, 5.3.6, 5.3.7.
Homework 4 (Due September 15, 2015) Problems from Bloch 2nd Edition. 6.2.1, 6.2.2, 6.3.2, 6.3.4, 6.3.5, 6.4.1, 6.4.4, 6.5.3, 6.5.5, 6.5.6.

Tests

Mid-Term I: September 15, 2015
Mid-Term II: October 15, 2015
Finals: December 2, 2015

Course Syllabus

Logic, proof techniques, sets, functions, relations, cardinality, order Topological spaces and continuous functions Connectedness, compactness, countability, separation axioms, metric topologies Measures, integration, signed measures

Course Description

This course is designed to enable engineering graduate students to learn basic mathematics for pursuing graduate level courses in mathematics. This course will provide the foundations in logic, introduction to topological and metric spaces, and probability measure.