Instructor: Dr. György ELEKES

Text: M. Laczkovich, Conjecture and Proof, the Fundamentals of Mathematical Thinking + parts of M.~Kac and S.~Ulam, Mathematics and Logic

Prerequisite: Introductory math courses

The purpose of this course is to give an introduction to mathematical thinking and to exhibit typical arguments and various methods of proofs. Problem solving is an integral part of the course.

Topics:

I. Proofs of impossibility, proofs of non-existence:

 Some typical examples of proofs of impossibility. Simple proofs of irrationality Ö ⁿ k, e,  p .
 The elements of the theory of geometric constructions. Fields. The impossibility of doubling the cube and trisecting all angles. Constructible regular polygons.
 Algebraic and transcendental numbers. e is transcendental. Approximation of irrationals by rationals. Liouville numbers.
 Hamel bases. Hilbert's third problem: a cube cannot be decomposed into polyhedra and reassembled to give a regular tetrahedron.

II. Constructions, proofs of existence:
 Cantor's proof for the existence of transcendental numbers. Countable sets. Sets of the power of the continuum.
 Equivalence of sets. The Cantor-Bernstein theorem.  Congruencies in R¹,  R² and R³. The Hausdorff paradox.
 Equivalence by finite decomposition. The Banach--Tarski paradox. Zermelo's non-measurable set. The Peano curve.
 Borel classes and universal sets. The existence of Borel sets of arbitrary class.
 A pure existence proof: the divisor game.