Jordan University of Science and Technology
Faculty of Science and Arts
Department of Mathematics and Statistics
Second Semester 2006-2007
Course Information
Course Title
Set Theory and Logic
Course Number
Math 245
Prerequisites
Math 102
Course Website
Instructor
Dr. Ahmed Al-Rawashdeh
Office Location
PH 2 level 1
Office Phone
23453
Office Hours
Sun., Tues., Thur., 10:15-11:15 and Mond., Wed., 12:45-2:15
E-mail
rahmed72@just.edu.jo
Teaching Assistant
Rasha Qawasmeh
Course Description
In this course we mainly study the following subjects Logic and Proofs, Set Theory, Relations,
Functions, Cardinality of Sets and Cardinal Numbers, The Axiom of Choice and Zorn’s Lemma, and
finally the Ordinal Numbers.
Text Book
Title
Transition to Advanced Mathematics
Author(s)
Publisher
D. Smith, M. Eggen and R. St. Andre
Books/ Cole Publishing Comp., Monterey, California.
Year
2001
Edition
Fifth
Book Website
References
1. Shwu-Yeng T. Lin and You-Feng Lin “Set Theory: An Intuitive
Approach”, First Edition, Houghton Mifflin Comp., Boston.
2. Introduction to Set Theory, Monk J. Donald, McGraw-Hill Inc. New
York.
3. Set Theory and Logic, Fraenkel Abraham A., Addison-Weseley,
Reading Mass.
4. Set Theory and Related Topics, Schaum’s Outline Series,
McGraw-Hill Book Company.
5. The Logic Book, M. Bergmann, J. Moore and J. Nelson, McGrawHill Book Company.
Assessment Policy
Assessment Type
First Exam
Expected Due Date
March
25th
29th
, 2007
Second Exam
April
Final Exam
June 4-6, 2007
, 2007
Weight
20%
20%
40%
Assignments
Biweekly
20%
Course Objectives
1.
Weights
To know the logical statements, connectives and the quantifiers.
15%
2.
To be familiar of all types of mathematical proofs: direct, indirect and proof by
contradiction. Also to prove statements using principal of mathematical induction
(PMI).
3. To study the sets: their notations, and operations. Also to study the indexed family
of sets, and to know how to work with arbitrary union and arbitrary intersection of
sets.To study relations: equivalence relations and its classes and partial order
relations, then to solve variety of problems concerning such relations.To define
partitions, and to see how it is connected with equivalence classes.To study
functions, its notations, and to define injective, surjective and bijective functions;
main properties and results.
20%
4. To study functions, its notations, and to define injective, surjective and bijective
functions; main properties and results. To know the induced sets functions: the
image and the inverse image of sets under a certain function.
5. To study the cardinality of sets, so we define equipotent of sets, finite sets,
denumerable sets, countable and uncountable sets, and then to classify sets up to
their cardinality.
6. To study the well ordering principal (WOP) and to prove that PMI is equivalent to
WOP. To know more definitions and results concerning the partially ordered sets
(posets): maximal (minimal) elements, Zorn’s lemma. To know the concepts of
ordinal numbers, and their operations.
20%
20%%
15%
10%
Teaching & Learning Methods
Learning Outcomes: Upon successful completion of this course, students will be able to
Related Objective(s)
Reference(s)
1
Chapter 2 and Handouts
1,2
1-6
3,4
5,6
6
5
Useful Resources
The university Library and the internet. References are good too.
Course Content
Week
Topics
Chapter in Text
(handouts)
3
4
5
6
7
8
9
10
11
12
13
1
Review. Logic: statements, connectives, tautology,
contradictions and equivalent statements.
2
Quantifiers: variety of examples, open
sentences
and
their
negations.
Mathematical proofs: direct, indirect and
proof by contradictions
1
1
Two
parts
proof,
proof
involving
quantifiers. Principal of mathematical
induction (PMI), principal of complete
induction and solving different examples.
1
The sets: power sets, union, intersection
and difference, universe sets and the
complements. Indexed family of sets:
arbitrary
union
and
arbitrary
intersection, pair-wise disjoint family
sets.
Generalizing
De’Morgan’s
laws.
Other forms of PMI by using the sets
notations
The well ordering principal of natural
numbers
(WOP),
proving
that
PMI
is
equivalent to WOP. Relations: Cartesian
product, the definition of relations, the
range and the domain of relations.
The inverse relation and the composition.
Equivalence
relations
and
equivalence
classes. The partial order relations,
partially ordered sets (Posets) and the
totally ordered sets.
Partitions: definition, and proving that a partition on a set
is equivalent to an equivalence relation on that set.
Functions: definitions (well-defined function), notations,
equality of functions, inverse function, composition of
functions. Types of functions: injective, surjective and
bijective.
Solving problems concerning bijective functions,
composition of bijective functions is a bijective function,
the existence of the inverse function.
The induced set functions: the image and the inverse
image of subsets under a certain function, the image
and the inverse image of an arbitrary union and
intersections, the image and the inverse image under a
bijective map, solving different problems.
Cardinality of sets: equipotent sets, equipotent as an
equivalence relations between sets. Finite sets: every
subset of finite is a finite, the finite union of finite sets is
also a finite set.
Denumerable and countable sets: the set of integers is
countable, the set of real numbers is uncountable, every
infinite subset of denumerable is denumerable, every
subset of countable is countable, the union of two
denumerable sets is denumerable.
The set of rational numbers is a countable set and the
irrational is uncountable. Cardinal numbers, the cardinal
number of denumerable sets, the cardinal number of the
real numbers (continuum number), Cantor’s theorem,
operations on cardinal numbers; their products, sums.
The exponential of cardinal numbers, the cardinal
number of denumerable sets is strictly less than the
continuum number. Continuum hypothesis and
continuum problem.
2
2, 3
3
3,4
4
4
5
5
5
5
14
The axiom of choice, totally ordered sets and chains, the
Hausdorff Maximality principal, Zorn’s lemma, The
ordinal numbers: definitions and operations. Review.
5
Additional Notes
Assignments
Will be given during classes
Exams
Cheating
Attendance
Against the university rules
Obligatory
Workload
Graded Exams
Exams (except the final) will be graded by the instructor, then return back to
students in a week
Participation
Many questions will be asked during the class, extra partial credits will be given
Laboratory
Projects