The aims of this course are to provide a basic introduction to various methods of proof used in mathematics and to the fundamental ideas in the study of sets, numbers and functions.
The language of mathematics. Mathematical statements, quantifiers, truth tables, proof.
Number theory I. Prime numbers, proof by contradiction
Proof by induction. Method and examples.
Set Theory. Sets, subsets, well known sets such as the integers, rational numbers, real numbers.Set Theoretic constructions such as unions, intersections, power sets, Cartesian products.
Functions. Definition of functions, examples, 1-1 and onto functions, bijective functions, composition of functions, inverse functions.
Counting. Counting of (mostly) finite sets, inclusion-exclusion principle, pigeonhole principle, binomial theorem.
Euclidean Algorithm. Greatest common divisor,proof of the Euclidean Algorithm and some consequences, using the Algorithm.
Congruence of Integers. Arithmetic properties of congruences,solving certain equations in integers.
Relations. Examples of various relations,reflexive, symmetric and transitive relations. Equivalence relations and equivalence classes. Partitions.
Number Theory II. Fundamental theorem of Arithmetic, Fermat’s little theorem.
Binary Operations. Definition and examples of binary operations. Definition of groups and fields with examples. Proving that integers mod p ( p a prime) give a finite field.