Numbers, Proofs and Counting

• Credit value: 30 credits at Level 4
• Convenor and tutor: Steve Noble
• Assessment: short, problem-based assignments and online tests (25%), a one-hour in-class test (25%) and a two-hour examination (50%)

This is a core Level 4 module on all BSc programmes involving mathematics. It covers key concepts of number systems, proof techniques, logic and mathematical reasoning, all of which are important preparation for more advanced modules.

Indicative module syllabus

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

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

• Definition and arithmetic with complex numbers
• The argand diagram
• Loci

Binary operations

• Definition and examples of binary operations
• The properties of commutativity and associativity
• Identity elements and inverses
• Idempotents
• Operation multiplication tables

Counting and probability

• Basic counting
• Product rule
• Strings from finite alphabets
• Counting subsets and binomial coefficients
• Ordered subsets and permutations
• Addition rule
• Naïve idea of probability triples
• Conditional probabilities
• Finding probabilities using counting results

By the end of this module, you will have:

• knowledge and understanding of, and the ability to use, mathematical and/or statistical techniques
• knowledge and understanding of a range of results in mathematics
• an appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments
• an understanding of the importance of assumptions and an awareness of where they are used and the possible consequences of their violation
• an ability to present, analyse and interpret data
• a deeper knowledge of some particular areas of mathematics, in particular, proof methods such as proof by induction and proof by contradiction; number sets and their properties, including integers, rational, real and complex numbers; modular congruences and their solutions, binary operations, counting and introductory probability.