Exercises in Modern Mathematics

This resource consists of ten topics, each section containing a brief explanation, examples and exercises.

Sets begins with a definition of a set and continues with the elements of a set, set notation, subsets, intersection and union of sets, Venn diagrams, the laws of sets and concludes with a number of problems.

Sets of points considers sets defined by algebraic rules, ordered pairs and moves on to graph lines and regions leading to solving simultaneous equations graphically and solving inequalities graphically. The section continues by looking at relations between sets and subsets leading to use of functions, mappings, composite functions, domain and range.

Linear programming begins with a detailed example of how linear programming can be used to solve problems leading to solving maximising and minimising problems.

Sets, Logic and Switching Circuits continues earlier work on sets before moving to consider a series of logic problems, switching diagrams, truth tables and Boolean algebra.

Scales of Notation begins by considering binary numbers, converting from denary to binary and performing addition, subtraction and multiplication in base two. Decimal and fraction conversion to binary is explored along with the use of a slide rule and index notation.

Groups describes the notion of a group and considers groups of symmetries, identities, inverses, groups in arithmetic and algebra and isomorphic groups.

Matrices starts with a basic definition followed by the idea of orders of matrices, transformation matrices, identity matrices, matrix multiplication, inverse matrices and continues to consider triple matrix products in geometry.

Vectors begins with simple vector addition, free vectors and applications in elementary geometry most notably in their use for proof. Further applications are considered such as the use of vectors to represent forces. The scalar product is described and used to find angles between vectors and use is made of unit vectors using i,j notation.

Probability and Statistics looks at how to collect data, represent data, calculate mode, median and mean, the dispersion of data including the interquartile range and standard deviation. Probability and chance contains simple examples of probability.

Topology begins by considering different categories of shapes, a Mobius strip and the number of faces edges and corners of solids leading to Euler’s formula. The section on networks includes the famous Bridges of Konigsberg problem and colouring networks.