Proof and Structure (EMMS095S)
This year I am teaching Proof and Structure.
This module introduces the essential logical and theoretical concepts necessary for the degree-level study of mathematics. At the end of the module you will have obtained a thorough grounding in the basic mathematical concepts which will be used throughout the degree programme.
What makes mathematics different from the sciences is the emphasis placed on proof. In this module you will be introduced to various common methods of proof, such as proof by contradiction and proof by induction, as well as the basic components of propositional logic: connectives, quantifiers and truth tables.
The main number sets (natural numbers, integers, rational, real and complex numbers) will be defined and their properties studied. Binary relations and binary operations and the concept of a group will be introduced and some elementary results established.
- The language of mathematics: statements; theorems; definitions; logical connectives (not, and, or, implies); truth tables; tautologies and equivalent statements; the universal and existential quantifiers; negating if/then statements and statements involving quantifiers; some elementary proofs; contrapositive proofs; converses and counterexamples; proof by contradiction; Euclid's proof of the infinity of primes; proof by induction; strong induction
- The number sets: integers, rationals and real numbers; the well ordering property; the division theorem; the fundamental theorem of arithmetic; the Euclidean division algorithm; algebraic properties of the natural numbers and integers; congruence and modular arithmetic; solution of linear congruences; the rational numbers, the real numbers; boundedness; the completeness axiom
- Complex numbers and infinite sequences: the complex numbers: arithmetic of complex numbers, polar form, De Moivre's Theorem and applications; definition of a sequence; limits and limit theorems; monotone sequences; subsequences; divergent sequences; infinite series
- Binary relations, binary operations and groups: binary relations and equivalence relations; binary operations; the properties of commutativity and associativity; identity elements and inverses; idempotents; operation multiplication tables; definition of a group; examples from geometry, permutations, matrices, number sets; cyclic groups and abelian groups; orders of elements and groups; subgroups; Lagrange's Theorem