Foundations of Mathematics

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Cursusdoel

After completing this course students are able to:
  1. demonstrate the various forms of mathematical proof
  2. use mathematical notation
  3. read and write a mathematical proof
  4. explain some fundamental notions and theorems of mathematics
Relationship between assessment and learning goals:
  1. Written in-class exam 1: This exam tests your ability to explain the meaning of, and use, the mathematical notation appearing in the first half of the course (course goals 2 and 4). Your ability to explain and prove some fundamental theorems will also be tested (course goals 1, 3, and 4).
  2. Written in-class exam 2: This exam tests your ability to explain the meaning of, and use, the mathematical notation appearing in the course (course goals 2 and 4). Focus will lie on the material from the second half of the course. Your ability to explain and prove some fundamental theorems will also be tested (course goals 1, 3, and 4).
  3. Homework assignments: These assignments will primarily test your ability to read and write a mathematical proof (course goal 3). They will also test your ability to use mathematical notation (course goal 2).

Vakinhoudelijk

This course introduces the students to academic mathematics. The big difference with high-school mathematics is its emphasis on proof. The student learns about logic and various forms of proof, such as the direct method, proof by contradiction and proof by complete induction.
These concepts will be applied to various fields of mathematics, such as set theory and number theory. Along the way, the student becomes acquainted with the language and notations of mathematics.
The course highlights the main attraction mathematics has for its practitioners: the joy of solving a puzzle. Every proof contains a sparkle of ingenuity, and there is great intellectual satisfaction in discovering the essential step in a proof, or admiring the brilliance of someone who found it before you. A typical problem is for instance the question whether the square root of 2 is a fraction. The answer came as a great shock to the ancient Greeks and its proof is both simple and very clever.
Another feature of the course is an introduction to the mysteries and paradoxes of the concept ʽinfinityʼ. Are there more real numbers than integers? (Yes.) Is the set of fractions larger than the set of integers? (No.)
Finally, there is a big emphasis on writing proofs. A proof should be logical, clear and do precisely what it should: convince a reader of the truth of some mathematical statement. Writing good proofs is a difficult art, which requires practice and the highest intellectual precision.

Format
There will be lecture sessions and exercise sessions. These last sessions form the most important part of the course, as the course is focused on students making their own mathematical discoveries and writing them down correctly. Students are required to regularly hand in exercises, which will be returned and can be handed in again after corrections. Because a mathematical proof is never ʽapproximatelyʼ right, it may take a few cycles of corrections/editing before such a hand-in is acceptable.


Werkvormen

UCU SCI 1 course

Toetsing

Mid-term Exam

Verplicht | Weging 40% | ECTS 3

Final Exam 1

Verplicht | Weging 40% | ECTS 3

Homework assignments

Verplicht | Weging 20% | ECTS 1,5

*midterm FEEDBACK*

Niet verplicht

Ingangseisen en voorkennis

Ingangseisen

Er is geen informatie over benodigde voorkennis bekend.

Voorkennis

Secondary school knowledge of mathematics is required. Passed final high school exams in IB Analysis & Approaches Higher Level Mathematics, Dutch VWO “Wiskunde B”, or similar “Calculus and Algebra” courses from a foreign high school should be sufficient. Students who do not meet this math requirement should take UCACCMAT01 first.

Voertalen

  • Engels

Competenties

  • Kritisch lezen

Cursusmomenten

Gerelateerde studies

Tentamens

Er is geen tentamenrooster beschikbaar voor deze cursus

Verplicht materiaal

Materiaal Omschrijving
BOEK Richard Hammack, Book of Proof, Edition 3, 2018. Book is available online for free. https://www.people.vcu.edu/~rhammack/BookOfProof/

Aanbevolen materiaal

Er is geen informatie over de aanbevolen literatuur bekend

Coördinator

dr. V.N.E. Blasjö v.n.e.blasjo@uu.nl

Docenten

dr. N.C. Taggart n.c.taggart@uu.nl

Inschrijving

Let op: deze cursus is niet toegankelijk voor studenten van andere faculteiten, bijvakkers mogen zich dus niet inschrijven.

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