Aims & Learning Objectives:
Aims: To introduce the concepts of logic that underlie all mathematical reasoning and the notions of set theory that provide a rigorous foundation for mathematics. A real life example of all this machinery at work will be given in the form of an introduction to the analysis of sequences of real numbers.
By the end of this course, the students will be able to: understand and work with a formal definition; determine whether straight-forward definitions of particular mappings etc. are correct; determine whether straight-forward operations are, or are not, commutative; read and understand fairly complicated statements expressing, with the use of quantifiers, convergence properties of sequences.
Logic: Definitions and Axioms. Predicates and relations. The meaning of the logical operators ∧, ∨,, →, ↔, ∀, ∃. Logical equivalence and logical consequence. Direct and indirect methods of proof. Proof by contradiction. Counter-examples. Analysis of statements using Semantic Tableaux. Definitions of proof and deduction.
Sets and Functions: Sets. Cardinality of finite sets. Countability and uncountability. Maxima and minima of finite sets, max (A) = - min (-A) etc. Unions, intersections, and/or statements and de Morgan's laws. Functions as rules, domain, co-domain, image. Injective (1-1), surjective (onto), bijective (1-1, onto) functions. Permutations as bijections. Functions and de Morgan's laws. Inverse functions and inverse images of sets. Relations and equivalence relations. Arithmetic mod p.
Sequences: Definition and numerous examples. Convergent sequences and their manipulation. Arithmetic of limits.