Foundations

Core course for B.Sc. (Research) Mathematics. Not available as UWE.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: None

Overview: Introduction to modern mathematical language and reasoning: Sets and Logic, Proof strategies, Functions, Induction.

Detailed Syllabus:

1. Sentential Logic: Deductive reasoning, negation of a sentence, conjunction and disjunction of sentences, equivalence of sentences, truth tables, logical connectives.
2. Sets: Operations on sets, Venn diagrams, cartesian product, quantifiers.
3. Proof Strategies: Direct proofs, proofs involving negations, conditionals, conjunctions, and disjunctions, existence and uniqueness proofs, proofs involving equivalence.
4. Relations and Functions: Ordered pairs, equivalence relations, equivalence classes, partitioning of a set, functions as many-one relations, graphs of functions, one-one functions, onto functions, inverse of a function, images and inverse images of sets.
5. Mathematical Induction: Division algorithm, principle of mathematical induction, well ordering principle, strong induction, principle of recursive definition.
6. More on Sets: Finite and infinite sets, countable and uncountable sets.

References:

1. Book of Proof by Richard Hammack, 2nd edition, Richard Hammack.
2. Mathematical Thinking by Keith Devlin, Lightning Source.
3. How to Prove It by Daniel J. Velleman, Cambridge University Press.
4. Mathematical Writing by Franco Vivaldi, Springer.
5. Proofs and Fundamentals by Ethan D. Bloch, Springer.
6. Introduction to Logic and to the Methodology of Deductive Sciences, Alfred Tarski, Oxford University Press.

Past Instructors: Amber Habib, Priyanka Grover