Course Title:                    Set Theory

Course Code:                   MATH 3274

Course Type:                    Core

Level:                                3

Number of Credits:            3

Semester:                          1

Prerequisite(s):                   MATH 2272 and MATH 2277

Course Rationale

Mathematics is an engaging field that provides powerful tools for all fields of applied science. Concepts arising from the study of abstract algebra are fundamental for all students and practitioners of mathematics. The central ideas and proofs emerging from this field have shaped human thought over the years. Those who have contributed to the development of this field are considered among the greatest thinkers of all times, and their achievements are held in high esteem.

This course serves to sharpen the analytical and critical reasoning skills of the student, and to improve his/her ability to express sophisticated mathematical ideas with clarity and coherence. An important objective of the course is to provide students with an opportunity to enhance their ability to abstract ideas, as this is one of the defining characteristics of any good mathematician. Particular emphasis will be placed on the powerful techniques and results that one obtains by investigating the structure of several mathematical objects in their full generality.

Students taking this course will be drilled in the practice of analytical reasoning, in order to obtain solutions and to interpret results. For this reason, this course is particularly relevant for students who intend to teach mathematics at the high school level. As the process of mathematical abstraction is central to the development of a mathematician, this is a core course for students pursuing the major in Mathematics.

Course Description

Students who take this course will require knowledge of the basic concepts of Algebra. They will also be required to have a solid grounding in elementary set theory and basic logic.  Thus, ABSTRACT ALGEBRA I is listed as a prerequisite.

The first part of the course involves axiomatic set theory, which includes philosophy of sets. The language of set theory is used to describe representations of relations and functions. A fundamental approach to concepts in set and the algebraic structures of groups, rings and fields is utilized to develop number systems. These systems include the natural numbers, integers, rationals, reals and complex numbers. The course proceeds onto a treatise on infinite sets and on the different cardinal numbers that lead to transfinite arithmetic. Axiom of Choice and its equivalent representations are then introduced, as well as point-set topology.

Since cogent communication of mathematical ideas is important in the presentation of proofs, the course will emphasize clear, concise exposition. This course will therefore be useful for all students who wish to improve their skills in mathematical proof and exposition, or who intend to study more advanced topics in mathematics.

Learning Outcomes

Upon successful completion of the course, students will be able to

• Define relations and functions, and provide and interpret their representations.
• Prove properties of the various number systems
• Compare and contrast the properties of numbers and number systems.
• Construct the rationals, reals and complex numbers using equivalence relations.
• Distinguish between the different types of cardinal numbers.
• Write standard proofs.

Course Content

1. The Axioms of Set Theory

The Axiom of Extension. The Axiom of Specification. The Axiom of Pairing. Unions and intersections. Complements and power sets. Families of sets.

2. Relations

Ordered pairs. The definition of a relation. Functions. Inverses and composites. Partial orders. Lower bounds and upper bounds. The infimum and the supremum.

3. The Natural Numbers

Successors. The construction of the natural numbers. The Peano Axioms. Arithmetic. The Principle of Mathematical Induction. The Recursion Theorem. The Well-ordering Principle.

4. The Real Numbers

The Least-Upper-Bound Property. Dedekind cuts. The construction of the real numbers.

5. Cardinality

The Schröder–Bernstein Theorem. Countable sets. Cantor’s Theorem. Uncountable sets and Cantor’s diagonal argument. The uncountability of the reals. Cardinal numbers. The basics of cardinal arithmetic. Continuum hypothesis.

6. Axiom of Choice

Axiom of Choice and its equivalent representations. Zorn’s Lemma. Well-ordering Principle (no proof of their equivalence).

7. Introduction to Point-Set Topology

Teaching Methodology

Lectures:  Three lectures each week (50 minute each).

Assessment

Coursework - Two coursework exams (20% each), Assignments (10%).   Total: 50%

Final Examination:  One 2-hour written paper.                                                  Total: 50%

Course Calendar

 WEEK TOPICS TUTORIAL & CW EXAM DISCUSSED TUTORIAL GIVEN 1 Introduction/Course Overview Fundamental concepts in set theory - 2 Philosophy of set relations 3 Functions, Relations and their representations Assignment 1 due. 4 Homomorphisms – Development of number systems 5 Properties of natural numbers Assignment 2 due. 6 Integers, rationals 1st coursework examination 7 Reals and complex number systems Assignment 3 due. 8 Infinite sets and cardinalities 9 Transfinite arithmetic Assignment 4 due. 10 Axiom of Choice 11 Introduction to Point-Set Topology Assignment 5 due. 12 Point-Set Topology continued 2nd coursework examination 13 Revision Assignment 6 due.

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Essential Texts:

• Math 2100 Lecture Notes – Edward Farrell, Department of Mathematics & Computer Science, 2nd edition, (2008).
• Solved Problems in Abstract Algebra – Edward Farrell, Department of Mathematics & Computer Science, (2009).
• Paul R. Halmos. Naïve Set Theory. Martino Publishing. 2011

Other Reference Texts:

Modern  Abstract Algebra – Frank Ayres, Mc Graw -Hill, , ISBN 978007002651, (2009).