Course Code:                 MATH 1152

Course Title:                  Sets and Number Systems

Level:                             1

Semester:                       1

No. of Credits:                 3

Pre-requisite(s):          Units 1 and 2 of CAPE Pure Mathematics, GCE A-Level Mathematics or equivalent.

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Course Rationale

Mathematics is a powerful tool for solving practical problems and is a highly creative field of study, combining logic and precision with intuition and imagination.  A basic knowledge of mathematics is needed to provide the necessary framework for solving problems in fields such as medicine, management, economics, government, computer science, physics, psychology, engineering, and the social sciences.

The ability to employ mathematical reasoning is a fundamental skill for any well-educated individual in the pure and applied sciences. This course serves to introduce science students to the basic concepts of algebra, rules of mathematical logic, set theory and number systems. It is an introductory level course and is designed to provide a solid foundation for students interested in further studies in mathematics. It is therefore a prerequisite for students who intend to take advanced level courses in Mathematics or Statistics.

Course Description

This course covers the essential rules of logic used for the formulation of mathematical proofs. Students are introduced to the basic operations that are performed on sets. In so doing the concepts of relations, functions and binary operations are examined via a set-   theoretic approach. A group is defined as a set with an operation.

Next we examine natural numbers and the derived principle of mathematical induction. We discuss permutations, combinations and sequences that are defined recursively with respect to natural numbers. The properties of the integers such as divisibility, greatest common divisor and Euclidean algorithm are examined. This continues to a section on number theory which includes infinitude of the primes fundamental theorem of Arithmetic and linear Diophantine equations. Next we study the field axioms in relation to rational numbers.  We solve linear and non-linear inequalities involving the real numbers. This extends to the triangle inequality, Arithmetic Mean – Geometric inequality and sums of infinite series with no tests for convergence. The basic numbers of complex numbers are discussed.

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through assigned problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and in the Help-Desk. Assessment will be continual and will be based on the assignments and in-course tests followed by a final examination based on the whole course.

The Help-Desk will be manned by teaching assistants who will give assistance (outside of the formal tutorials) to students on a one-on-one basis or in small groups

How To Study For This Course

“There is no royal road to Geometry” response by Euclid to King Ptolemy’s request for a quick introduction to geometry.

Mathematics is not a spectator sport. In order to develop understanding and problem solving skills it is essential that students do many exercises and problems, including many problems additional to those set in assignments. This is the best way to learn the material properly and also to revise. One cannot learn Mathematics by simply reading the notes given in class and looking over worked examples.

Mathematics and Actuarial Science majors will need to know and understand this material thoroughly and will use it in future courses. It must become firmly entrenched in long-term memory, not something to be forgotten the day after the final examination. Again, the best way to do this is through a great deal of practice in solving problems.

The Logic module in this course is relevant to future work in Mathematics but is also a useful skill in life. Try to construct examples within Mathematics and from other areas.

In addition to your lectures, there is a compulsory tutorial. It is very important that you attend and participate. Do not go just to `get the notes’. The purpose of the tutorial is to assist you in understanding what was done in the lecture. The Department also runs a Help Desk. There you may get one-on-one assistance from Tutors. Check the Department or MyeLearning for details.

Many students buy textbooks but never use them. They try to rely only on lecture notes. Read your Textbook! Do not be solely dependent on your lecturer. Textbooks give you a second perspective on the material covered in the lecture. They are also valuable sources of additional problems and exercises.

Learning Outcomes

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

• Analyse arguments via the rules of logic.
• Formulate mathematical proofs by using the rules of logic.
• Prove that two sets are equal.
• Test for the basic properties of relations and functions.
• Illustrate the principle of mathematical induction.
• Distinguish between permutations and combinations.
• Interpret sequences of natural numbers that can be defined recursively.
• Test for some basic properties of real numbers.
• State the field axioms in the construction of rational numbers.
• Distinguish between real and natural numbers.
• Solve inequalities involving quotients and absolute values.
• Interpret the Arithmetic Mean-Geometric Mean Inequality.
• Calculate sums of simple infinite series without performing tests for convergence.
• Perform operations with complex numbers.

Course Content

Logic and Set Theory:

• Statements in Mathematics. Manipulation of statements: negation, conjunction, disjunction and implication. The notion of a set as the fundamental structure of Mathematics. Equality of sets. Subsets. Unions and intersections. Venn Diagrams.
• Illustration of logical statements, proof and validity of arguments using sets. Set Algebra and De Morgan’s Laws.

Relations:

• The Cartesian Product of sets.
• Definition of a function. Injective and surjective functions. The inverse of a function.
• A relation as a generalization of a function. Properties of relations. Reflexive, symmetric and transitive relations. Equivalence relations and partitions of sets.
• Binary operations. Properties of binary operations. Commutative, associative and distributive operations, existence of an identity and inverse.
• The definition of a group.

The Natural Numbers:

• The Principle of Mathematical Induction.
• Permutations and combinations.
• Simple sequences of natural numbers. Sequences defined recursively.

The Integers:

Basic facts regarding divisibility. The greatest common divisor and the Euclidean Algorithm. The infinitude of the primes. The Fundamental Theorem of Arithmetic. Linear Diophantine Equations.

The Rational Numbers:

The Field Axioms. The proof that  cannot be rational

The Real Numbers:

• Solution of linear and non-linear inequalities in R. The absolute value.

The Triangle Inequality.

• The Arithmetic Mean-Geometric Mean Inequality.
• Sums of simple infinite series of real numbers (no tests for convergence).

The Complex Numbers:

Real and imaginary parts of a complex number. Complex conjugates. The modulus and

argument of a complex number. The polar and exponential forms of a complex number.

The Triangle inequality.

Teaching Methodology

Lectures: Two (2) lectures each week fifty (50) minutes each.

The primary mode of instruction is the face-to-face lecture. Lecturers vary in how they present the material but most will use presentation tools (multi-media) in presentation of their lectures.

Tutorials: One (1) tutorial session for eleven weeks, fifty (50) minutes each. These tutorials will serve as a forum for discussing concrete examples that students will encounter in their respective fields. Difficulties in the assignments will be discussed there. You may also be required to solve problems during class and to make presentations.

MyeLearning

myeLearning is a valuable site for all students and is the main communication tool between students and lecturer. Most of the course materials and information will be found there.

Assessment

Final Examination (one 2-hour written paper) - 50%

Coursework: two in-course tests (40%) and assignments (10 %).

You will be given one assignment per week (a total of 10), except when tests are due, all of which will be graded. Nine assignments will count towards the total mark.

The final examination consists of essay type questions involving definitions, statements of theorems, proofs of major theorems and problems. The coursework examination consists of shorter essay type questions which focus more on knowledge and understanding and solving simpler problems. The final examination will place greater emphasis on problem solving skills. The assignments consist of problems based on lectures that are covered in each week. These questions are compulsory. They are given out each week, as indicated in the course calendar, whenever there is no coursework examination.

Course Calendar

Week

Lecture subjects

Tutorials

1

Introduction /Course Overview

Logic and Set Theory: Statements in Mathematics. Manipulation of statements. Negation, conjunction, disjunction and implication. The notion of a set as the fundamental structure of Mathematics.

Assignment 1 given.

2

Logic and Set Theory:

Equality of sets. Subsets. Unions and Intersections. Venn Diagrams. Illustration of logical statements. Proof and validity of arguments using sets. Set Algebra and De Morgan’s Laws.

Assignment 2 given, Assignment 1 due.

3

Relations: The Cartesian Product of sets. Definition of a function. Injectivity and Surjectivity. The inverse of a function. A relation as a generalistion of a function. Reflexive, symmetric and transitive properties of relations.

Assignment 3 given, Assignment 2 due.

4

Relations. Equivalence relations and partitions of sets. Binary operations and their properties. Commutativity, associativity, distributivity, existence of an identity and of inverses. The definition of a group.

Assignment 4 given, Assignment 3 due.

5

The Natural Numbers: The Principle of Mathematical Induction. Permutations and Combinations.

Assignment 5 given, Assignment 4 due.

6

The Natural Numbers: Simple sequences of natural numbers. Sequences defined recursively.

Coursework Exam #1 (15%)

7

The Integers:  Basic facts regarding divisibility. The greatest common divisor and the Euclidean Algorithm.

Assignment 6 given, Assignment 5 due.

8

The Integers: The infinitude of the primes. The Fundamental Theorem of Arithmetic. Linear Diophantine Equations.

Assignment 7 given, Assignment 6 due.

9

The Rational Numbers/The Real Numbers : The Field Axioms. Proof that   is not rational. Solution of linear and non-linear inequalities in R.

Assignment 8 given, Assignment 7 due.

10

The Real Numbers: The absolute value. The Triangle Inequality. The Arithmetic Mean- Geometric Mean Inequality. Sums of simple infinite series of real numbers ( no tests for convergence) .

Assignment 9 given, Assignment 8 due.

11

The Complex Numbers: Real and Imaginary parts of a complex number. Complex conjugates. The modulus and argument of a complex number.

Assignment 10 given, Assignment 9 due.

12

The Complex Numbers: The polar and exponential forms of a complex number. The Triangle Inequality.

Coursework 2

(15%)

13

Revision

None

Essential Texts:

• Edward Farrell. Math 1140 Lecture Notes, Department of Mathematics & Statistics, 2nd edition (2008).
• Seymour Lipschutz. 2000 Solved Problems in Discrete Mathematics, (2009).

Other Suggested Texts / References

• Seymour Lipschutz. Theory and Problems of Set Theory and Related Topics –Schaums Outline Series, ISBN – 0 -07 – 038031 – 7 , (2009).