Update pure math M208 and M303 parts

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@@ -159,6 +159,88 @@ Using Minitab to carry out straightforward data analyses
 
 http://www.open.ac.uk/courses/qualifications/details/m208?orig=q31
 
+#### Unit A1 Sets, functions and vectors 
+Revises important foundations of pure mathematics and the mathematical
+language used to describe them.
+#### Unit A2 Number systems 
+Systems of numbers most widely used in mathematics: the integers, rational
+numbers, real numbers, complex numbers and modular or 'clock' arithmetic,
+and looks at when and how certain types of equations can be solved in the
+system.
+#### Unit A3 Mathematical language and proof
+Writing of pure mathematics and some of the methods used to construct
+proofs, and as a further introduction to abstract mathematical thinking
+equivalence relations are introduced.
+#### Unit A4 Real functions, graphs and conics
+Reminder of the principles underlying the sketching of graphs of functions
+and other curves.
+#### Unit B1 Symmetry and groups Symmetry of plane figures and solids, and shows how this topic leads to the
+definition of a group, which is a set of elements that can be combined with
+each other in a way that has four basic properties called group axioms.
+#### Unit B2 Subgroups and isomorphisms Subgroups, which are groups that lie inside other groups, and also at cyclic
+groups, which are groups whose elements can all be obtained by repeatedly
+combining a single element with itself. It also investigates groups that
+appear different but have identical structures.
+#### Unit B3 Permutations 
+Functions that rearrange the elements of a set: it shows how these
+functions form groups and looks at some of their properties.
+#### Unit B4 Lagrange's Theorem and small groups
+Fundamental theorem about groups, and uses it to investigate the
+structures of groups that have only a few elements, before focusing on
+improving skills in understanding theorems and proofs in the context of
+group theory.
+#### Unit C1 Linear equations and matrices 
+Why simultaneous equations may have different numbers of solutions, and
+also explains the use of matrices.
+#### Unit C2 Vector spaces 
+Generalises the plane and three-dimensional space, providing a common
+structure for studying seemingly different problems.
+#### Unit C3 Linear transformations 
+Mappings between vector spaces that preserve many geometric and
+algebraic properties.
+#### Unit C4 Eigenvectors 
+Diagonal representation of a linear transformation, and applications to
+conics and quadric surfaces.
+#### Unit D1 Numbers Real numbers as decimals, rational and irrational numbers, and goes on to
+show how to manipulate inequalities between real numbers.
+#### Unit D2 Sequences The 'null sequence' approach, used to make rigorous the idea of
+convergence of sequences, leading to the definitions of pi and e.
+#### Unit D3 Series 
+Convergence of series of real numbers and the use of series to define the
+exponential function.
+#### Unit D4 Continuity 
+Sequential definition of continuity, some key properties of continuous
+functions, and their applications.
+#### Unit E1 Cosets and normal subgroups 
+Revision of Units B1-B4 and looks at how a group can be split into 'shifts' of
+any one of its subgroups.
+#### Unit E2 Quotient groups and conjugacy 
+How to 'divide' a group by one of its subgroups to obtain another group, and
+how in any group some elements and some subgroups are similar to each
+other in a particular sense.
+#### Unit E3 Homomorphisms 
+Functions that map groups to other groups in a way that respects at least
+some of the structure of the groups.
+#### Unit E4 Group actions 
+How group elements can sometimes be applied to elements of other sets in
+natural ways. This leads to a method of counting how many different
+objects there are of certain types, such as how many different coloured
+cubes can be produced if their faces can be painted any of three different
+colours.
+#### Unit F1 Limits 
+The epsilon-delta approach to limits and continuity, and relates these to the
+sequential approach to limits of functions.
+#### Unit F2 Differentiation 
+Differentiable functions and gives L'Hopital's rule for evaluating limits.  
+Integration explains the fundamental theorem of calculus, the Maclaurin
+integral test and Stirling's formula.
+#### Unit F3 Integration 
+The fundamental theorem of calculus, the Maclaurin integral test and
+Stirling's formula.
+#### Unit F4 Power series 
+Finding power series representations of functions, their properties and
+applications.
+
 
 ### DD209 Running the economy
 http://www.open.ac.uk/courses/qualifications/details/dd209?orig=q15
@@ -177,6 +259,108 @@ http://www.open.ac.uk/courses/qualifications/details/m249?orig=q36
 http://www.open.ac.uk/courses/qualifications/details/m269?orig=r38
 ### M303   Further pure mathematics
 http://www.open.ac.uk/courses/qualifications/details/m303?orig=q31
+
+#### Chapter 1 Foundations 
+Proof by induction, divisibility, linear Diophantine equations
+#### Chapter 2 Prime numbers 
+Prime numbers, Fundamental Theorem of Arithmetic, prime
+decomposition of integers, the theta-function, Fibonacci numbers.
+#### Chapter 3 Congruence 
+Definition of congruence, properties of congruence, divisibility
+tests, linear congruences, solution of linear congruences,
+solving systems of linear congruences.
+#### Chapter 4 Fermat's and Wilson's Theorems
+Fermat's Little Theorem, Wilson's Theorem, polynomial
+congruences, Lagrange's Theorem (for numbers)
+#### Chapter 5 Examples of groups 
+Group axioms, subgroups, cosets, Lagrange's Theorem (for
+groups), normal subgroups, quotient groups, conjugate
+elements, homomorphism of groups, isomorphism of groups,
+first isomorphism theorem, correspondence theorem.
+#### Chapter 6 Towards classification 
+Direct product of groups, internal direct product theorem,
+cyclic groups, direct product of cyclic groups, decomposition
+of finite cyclic groups, group actions, orbits and stabilisers.
+#### Chapter 7 Finite groups 
+Group presentations, dihedral groups, dicyclic groups,
+#### Chapter 8 The Sylow Theorems 
+Sylow p-subgroup, the Sylow theorems, applications of the
+Sylow theorems, prime power subgroups theorem.
+#### Chapter 9 Multiplicative functions 
+Multiplicative functions, Euler's phi-function, reduced set of
+residues, Euler's theorem, primitive roots.
+#### Chapter 10 Quadratic reciprocity 
+Solutions of quadratic congruences, quadratic residues,
+Euler's criterion, the Legendre symbol, Gauss's Lemma,
+quadratic character of 2, the law of quadratic reciprocity,
+quadratic character of 3, the Jacobi symbol.
+#### Chapter 11 Rings and polynomials 
+Ring axioms, subrings, units, fields, polynomials over fields,
+division algorithm for polynomials, factors of a polynomial,
+Euclidean algorithm for polynomials, factorising polynomials,
+irreducibility for polynomials, rational root test, Gauss's
+lemma, Eisenstein's criterion
+#### Chapter 12 Fermat's Last Theorem and unique factorisation
+Pythagorean triples; integral domains; associates,
+irreducibles and primes in rings; integral domains, norms for
+integral domains; Euclidean domains; division algorithm for
+Euclidean domains; highest common factors in Euclidean
+domains; unique factorisation domains.
+#### Chapter 13 Distance and continuity 
+Sequences in the real line; real null sequences; continuity of
+real-valued functions; intermediate value theorem; extreme
+value theorem; continuity on the plane; Euclidean distance on
+the plane.
+#### Chapter 14 Metric spaces and continuity 1 
+Continuity of functions from R^n to R^m, Euclidean distance on
+R^n, convergent sequences in R^n, metrics, metric spaces,
+convergence of sequences in metric spaces, continuity in
+metric spaces.
+#### Chapter 15 Metric spaces and continuity 2 
+Induced metrics, Cantor metric, equivalent metrics, product
+metrics, pointwise convergence of functions, uniform
+convergence of functions, the max metric on C[0,1].
+#### Chapter 16 Open and closed sets 
+Open sets, closed sets, dense sets, nowhere dense sets,
+closure of a set, interior of a set, boundary of a set, countable
+sets, uncountable sets.
+##### Chapter 17 Rings and homomorphisms 
+Fields of fractions, ring isomorphisms, primitive polynomials,
+ideals, principal ideals, principal ideal domain, algebra of
+ideals, cosets of an ideal, quotient rings, ring
+homomorphisms, maximal ideals, prime ideals.
+#### Chapter 18 Fields and polynomials 
+Isomorphism of fields, field extensions, vector spaces over
+fields, degree of a field extension, algebraic and
+transcendental elements, minimal polynomials, the KLM Theorem for field extensions, finite fields, roots of unity,
+splitting fields, splitting polynomials, classification of finite
+fields.
+#### Chapter 19 Fields and geometry 
+Subfield generated by a set, field extensions of finite degree,
+field of algebraic numbers, transcendental extension, ruler
+and compass constructions, constructible number,
+impossibility of doubling the cube, squaring the circle and
+trisecting the angle pi/3
+#### Chapter 20 Public-key cryptography 
+RSA cryptosystem, Diffie-Hellman cryptosystem, elliptic
+curves, Diffie-Hellman-ElGamal cryptosystem, Menezes-Vanstone cryptosystem.
+#### Chapter 21 Connectedness 
+Homeomorphisms, disconnections, connectedness,
+connected components, totally disconnected sets,
+connectedness in Euclidean spaces, the intermediate value
+theorem, path-connectedness, the topologist's cosine.
+#### Chapter 22 Compactness 
+Sequential compactness, the Heine-Borel theorem,
+generalised extreme value theorem, Arzela-Ascoli Theorem,
+open covers, compact metric spaces, equivalence of
+sequential compactness and compactness in metric spaces.
+#### Chapter 23 Completeness 
+Cauchy sequences, complete metric spaces, the contraction
+mapping theorem, completion of a metric space.
+#### Chapter 24 Fractals 
+The Hausdorff metric, self-similar sets, iterated function
+schemes, box dimension, open set condition.
+
 ### DD309  Doing economics: people, markets and policy
 http://www.open.ac.uk/courses/qualifications/details/dd309?orig=q15
 ### MST326 Mathematical methods and fluid mechanics