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
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.
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
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.
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
Examples of direct and indirect proofs and proof by cases.
Proof by contradiction, examples.
9 November - Dean's Day, no class.
16 November - mid term test.
Review of test questions.
Review of some questions from Mathematical Analysis test.
Recursive definitions (exponentiation, factorial). Sequences defined by recurrences.
No class. Exercise sheet on mathematical induction.