SYLLABUS
COURSE TITLE
FACULTY/INSTITUTE
INTRODUCTION TO LOGIC AND SET THEORY
FACULTY OF MATHEMATICS AND NATURAL
SCIENCES
COURSE CODE
DEGREE PROGRAMME
FIELD OF STUDY
DEGREE LEVEL
MATHEMATICS
FORMA
MODE
STUDIÓW/STUDY
FIRST DEGREE
FULL-TIME
BASIC
YEAR 1, SEMESTER 1 OR SEMESTER 2
URSZULA BENTKOWSKA, PHD
COURSE FORMAT
YEAR AND SEMESTER
NAME OF THE TEACHER
COURSE OBJECTIVES
1. GETTING TO KNOW THE CONCEPTS OF SENTENTIAL AND PREDICATE CALCULUS.
2. GETTING TO KNOW THE CONCEPT OF MATHEMATICAL INDUCTION.
3. GETTING TO KNOW THE BASIC CONCEPTS OF SET AND RELATION THEORY.
4. PRESENTATION OF THE BASIC CONCEPTS CONCERNING EQUIVALENCE RELATIONS
AND ORDERS.
5. PRESENTATION OF THE BASIC ISSUES CONCERNING THE POWER OF SETS.
PREREQUISITES
The knowledge of elementary mathematics on the level of
secondary school.
KNOWLEDGE:
LEARNING OUTCOMES
-
understanding the role and importance of proof in
mathematics, as well as the concept of the
importance of assumptions in the proof
knowledge of the concepts and methods of
mathematical logic, set theory, relation calculus, and
concepts concerning functions which are included in
the fundamentals of various disciplines of
mathematics
SKILLS:
-
uses the propositional and predicate calculus; able to
correctly use quantifiers also in everyday language
able in an understandable way (in speech and in
writing) to provide the correct mathematical
-
reasoning, formulate theorems and definitions
able to prove with the use of mathematical induction
and to define functions and relations recursively
uses the language of set theory, interpreting issues in
different areas of mathematics
an create new objects by constructing the quotient
sets
understand the issues associated with different types
of infinity and orders in sets
FINAL COURSE OUTPUT - SOCIAL COMPETENCES
knows the limits of his own knowledge and
understands the need for further education
COURSE ORGANISATION –LEARNING FORMAT AND NUMBER OF HOURS
Individual meeting with the teacher – 30 hours
COURSE DESCRIPTION
1. Propositional calculus.
Logical sentence. Chart of the propositional condition. Logical connectives.
Tautologies. The rules of proof construction.
2. Predicate calculus.
Tautologies of the predicate calculus. Examples of applications of tautologies.
3. Set theory.
Sets. Operations on sets. Generalized operations on sets.
4. Natural numbers.
The Peano axioms. Mathematical Induction. Recursion.
5. Relation calculus.
The Cartesian product of sets. Relations. Operations on relations. Classification of
relations.
6. Functions.
Property of functions. Images and inverse images of sets of functions.
7. Equivalence relations.
Equivalence relation. Class of abstraction and quotient set. The principle of
abstraction. Constructions of sets with the use of equivalence relations.
8. Power of sets.
Finite and infinite sets. Equinumerosity of sets. Countable and uncountable sets.
Cardinal numbers.
9. Orders in sets.
Partial order. Linear order. Types of linear orders. Kuratowski-Zorn's Lemma. The
axiom of choice.
METHODS OF INSTRUCTION
Solving tasks. Individual work.
REQUIREMENTS AND ASSESSMENTS
Solving tasks in writing.
GRADING SYSTEM
The student receives points for the solved tasks. Let S
be the sum of the points, then:
[0,50%S] – 2.0 (F)
(50%S,60%S] - 3.0 (E)
(60%S,70%S] - 3.5 (D)
(70%S,80%S] - 4.0 (C)
(80%S,90%S] - 4.5 (B)
(90%S,100%S] - 5.0 (A)
TOTAL STUDENT WORKLOAD
NEEDED TO ACHIEVE EXPECTED
LEARNING OUTCOMES EXPRESSED
IN TIME AND ECTS CREDIT POINTS
LANGUAGE OF INSTRUCTION
INTERNSHIP
MATERIALS
150 HOURS - 6 ECTS
ENGLISH
NOT APPLICABLE
PRIMARY OR REQUIRED BOOKS/READINGS:
U. Daepp, P. Gorkin, Reading, Writing, and Proving. A
Closer Look at Mathematics (Undergraduate Texts in
Mathematics), Springer, Bucknell University, 2011.
SUPPLEMENTAL OR OPTIONAL BOOKS/READINGS:
J. Słupecki, L. Borkowski, Elements of Mathematical
Logic and Set Theory, Pergamon Press, PWN-Polish
Scientific Publishers, Warszawa, 1967.