Course objectives:
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The aim of this course is to present basic notions, symbolic language and standard methods of mathematics. Students will learn the fundamentals of mathematical logic, especially propositional calculus and predicate logic, and they will practice work with compound statements, and statements containing quantifiers. Students will get acquainted with the logical structure of mathematics and its development from axioms to theorems. The basic types of mathematical proofs will be discussed in detail. The importance of examples and counterexamples in the process of building a mathematical theory will be mentioned. Also the importance of human intuition, sketch (and/or image) and experiment in the development of a rigorous scientific theory will be discussed. Possible modes of use of computers in mathematics as computational tool, experimental device as well as automated theorem proving will be explained in detail. The fundamental difference between these three modes will be illustrated and clarified by examples.
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Requirements on student
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To be able to read mathematical text; to be able to perform operations on logical expressions;
to be able to use basic types of proofs; to be able to localize invalid argument in a "proof".
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Content
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Week 1: Introduction; necessity of formalisation in mathematics; examples and thorough study of misleading and invalid arguments;
Week 2: Propositional calculus;
Week 3: Predicate logic - an introduction;
Week 4: Predicate logic - advanced examples;
Week 5: Basic types of proofs and its principles; Sets; Real Numbers
Week 6: Basic types of proofs by examples;
Week 7: Selected theorems of mathematical analysis and algebra under microscope, revealing their structure;
Week 8: Thorough analysis of selected proofs from other mathematical subjects;
Week 9: Logical structure of a mathematical theory;
Week 10: Exercise: a simple mathematical theory build from the ground up;
Week 11: Diskussion on intuition and experiment in mathematics; publishing and scientific ethic;
Week 12: Computer simmulations and experiments in mathematics; experimental mathematics;
Week 13: Rigorous computer (and/or computer assisted) proofs.
see also http://analyza.kma.zcu.cz.
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Activities
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Fields of study
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Guarantors and lecturers
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Literature
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Recommended:
R.M. Smulyan. Jak se jmenuje tahle knížka. Praha, 1986.
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Recommended:
R. Thiele. Matematické důkazy. SNTL, Praha, 1986.
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Recommended:
J. Polák. Přehled středoškolské matematiky. Prometheus, Praha, 2000.
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On-line library catalogues
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Contact hours
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26
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Preparation for comprehensive test (10-40)
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16
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Preparation for formative assessments (2-20)
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10
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Total
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52
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
ovládat základní matematické znalosti v rozsahu učiva střední školy |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
ovládat základní matematické dovednosti v rozsahu učiva střední školy |
Competences - students are expected to possess the following competences before the course commences to finish it successfully: |
N/A |
N/A |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
rozumět základním pojmům výrokové logiky |
rozumět základním pojmům predikátové logiky |
ovládat základní pojmy teorie množin |
popsat a aplikovat základní typy matematických důkazů |
na základní úrovni používat a citovat odbornou literaturu a mít základní povědomí o tom, co je plagiátorství |
Skills - skills resulting from the course: |
číst a porozumět matematickému textu (s kvantifikátory) |
umět pracovat s logickými výroky |
chápat rozdíly mezi axiomem, definicí, větou a hypotézou |
umět používat základní typy důkazových technik |
umět vyhledávat informace v MathSciNet a Scopus |
Competences - competences resulting from the course: |
N/A |
N/A |
N/A |
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Test |
Skills demonstration during practicum |
Skills - skills achieved by taking this course are verified by the following means: |
Test |
Skills demonstration during practicum |
Competences - competence achieved by taking this course are verified by the following means: |
Test |
Skills demonstration during practicum |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Practicum |
Skills - the following training methods are used to achieve the required skills: |
Practicum |
Competences - the following training methods are used to achieve the required competences: |
Practicum |
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