| Code |
13900
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| Year |
1
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| Semester |
S1
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Not applicable.
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Learning outcomes |
On completion of this unit successful students will be able:
1. to understand the meaning of mathematical statements involving quantifiers and logical connectives.
2. to construct and interpret truth tables for logical propositions.
3. to recognize incorrect and sloppy mathematical reasoning.
4. to construct and write elementary proofs of mathematical statements using a range of fundamental and standard proof techniques (direct argumentation, induction, contradiction, contraposition).
5. to use basic set-theoretic language and constructions to prove results about finite, denumerable and uncountable sets.
6. to understand the significance and the implications of the choice axiom.
7. to explain the structure of the number systems, both intuitively and axiomatically, especially Peano's axioms for the natural numbers and Dedekind's construction of the reals
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Syllabus |
1. Basic Logic: propositions and predicates, the elementary connectives, truth tables, quantifiers; arguments, premises, conclusions, truth and validity, natural deduction.
2. Methods of Proof: Direct proof, proof by contradiction, contrapositives; the induction principle and proof by induction; proof by cases.
3. Elements of Set Theory: operations on sets, relations, and functions, cardinality; a brief introduction to the axiomatic set theory (Russel’s paradox, the Axiom of Choice, well Ordering. the Continuum Hypothesis, Zorn’s Lemma)
4. Number Systems: natural numbers (Peano’s axioms), integer numbers, rational numbers, real numbers (axiomatic, Dedekind’s construction, construction via Cauchy’s sequences), the complex numbers, quaternions, and the Cayley numbers.
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Main Bibliography |
1. S. G. Krantz, The Elements of Advanced Mathematics, 4th edition, CRC Press, 2017
2. A. Franco de Oliveira, Lógica e Aritmética, Gradiva, 1991
3. P.J. Eccles, An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, Cambridge University Press,1997
4. Devlin Keith, Sets, functions, and logic, Chapman and Hall/CRC, 3rd edition, 2003
5. Daniel J. Velleman, How To Prove It. A Structured Approach, 2nd edition, Cambridge University Press, 2006
6. Elon Lages Lima, Curso de Análise - Volume 1, 14.ª Edição, Projeto Euclides, IMPA, 2014
7. M. T. F. de Oliveira Martins, Tópicos Fundamentais de Matemática, Departamento de Matemática, Faculdade de Ciências e Tecnologia da Universidade de Coimbra, 1999
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Language |
Portuguese. Tutorial support is available in English.
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