Classes for the course Mathematical Skills in the winter semester of 2015 give an introduction to propositional logic and predicate first-order logic, proof techniques such as mathematical induction and proof by contradiction, and a variety of example problems that illustrate different aspects of mathematical proof. Classes also enable students to consolidate their learning and mathematical background to help follow the Mathematical Analysis I and Discrete Mathematics I courses.

Mathematical Skills is an optional course, although credits are awarded for regular participation in classes and performance on the two tests that will be set, one mid semester, the other at its close.

Classes are held at 12:20 on Mondays (excepting holidays) in Room S11 in the MFF building at Mala Strana.

Material covered

5 October

Propositional calculus, atomic and compound propositions, truth tables, material implication ("if...then" and other formulations in natural English), De Morgan's rule for negation, conjunctive and disjunctive normal form

12 October

More versions of "P implies Q" in natural English (P is sufficient for Q, Q is a necessary condition for P, etc.) Tautologies and deductive arguments: direct proofs (modus ponens) and indirect proofs (modus tollens), corresponding to proving P implies Q or its contrapostive not Q implies not P.

19 October

Review of disjunctive normal form and use of truth tables in solving logic problems. Exclusive or as binary addition (odd number of disjuncts true). Universal and existential quantifiers: negation and importance of order (e.g. there is y such that x+y=0 for all x is very different to for all x there is y such that x+y=0, the latter expressing existence of an additive inverse). Direct and indirect proof schema. Proof by contradiction (e.g. Euclid's proof of infinitude of primes).

For more about logic see e.g. A.G. Hamilton, Logic for Mathematicians, revised ed. 1988

26 October

Examples of direct and indirect proofs and proof by cases.

2 November

Proof by contradiction, examples.

9 November - Dean's Day, no class.

16 November - mid term test.

23 November

Review of test questions.

30 November

Review of some questions from Mathematical Analysis test.

Recursive definitions (exponentiation, factorial). Sequences defined by recurrences.

7 December

No class. Exercise sheet on mathematical induction.