M208_5 Pure mathematics

Mathematical language

About this free course

This free course is an adapted extract from the Open Unviersity course M208: Pure Mathematics www3.open.ac.uk/study/undergraduate/course/m208.htm

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www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/mathematical-language/content-section-0.

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978-1-4730-1382-7 (.kdl)
978-1-4730-0614-0 (.epub)

Contents

- Introduction

- Learning outcomes

- 1 Sets

- 2 Functions

- 3 The language of proof

- 4 Two identities

- 5 Solutions to the exercises

- Conclusion

- Keep on learning

- Acknowledgements

Introduction

When we try to use ordinary language to explore mathematics, the words involved may not have a precise meaning, or may have
more than one meaning. Many words have meanings that evolve as people adapt their understanding of them to accord with new
experiences and new ideas. At any given time, one person's interpretation of language may differ from another person's interpretation,
and this can lead to misunderstandings and confusion.

In mathematics we try to avoid these difficulties by expressing our thoughts in terms of well-defined mathematical objects.
These objects can be anything from numbers and geometrical shapes to more complicated objects, usually constructed from numbers,
points and functions. We discuss these objects using precise language which should be interpreted in the same way by everyone.
In this course we introduce the basic mathematical language needed to express a range of mathematical concepts.

Please note that this course is presented through a series of downloadable PDF files.

This OpenLearn course is an adapted extract from the Open Unviersity course M208: Pure Mathematics

Learning outcomes

After studying this course, you should be able to:

- use set notation

- determine whether two given sets are equal and whether one given set is a subset of another

- find the union, intersection and difference of two given sets

- determine the image of a given function

- determine whether a given function is one-one and/or onto.

1 Sets

In Section 1 we discuss the idea of a set and describe some ways to define sets. We illustrate our discussion with sets of numbers and with geometrical sets of points
in the plane. We also explain how to check whether two given sets are equal and whether one set is a subset of another. Finally,
we introduce the set operations of union, intersection and difference.

Click the link below to open Section 1 (16 pages, 389KB).

Section 1

2 Functions

In Section 2 we give the general definition of a function, and illustrate how functions can be used to describe a variety of mathematical concepts, such as transformations of the
plane. We discuss the idea of composing two functions, and the idea of forming the inverse of a function.

Click the link below to open Section 2 (16 pages, 366KB).

Section 2

3 The language of proof

In Section 3 we examine the language used to express mathematical statements and proofs, and discuss various techniques for
proving that a mathematical statement is true. These techniques include direct proof, proof by mathematical induction, proof by contradiction and proof by contraposition. We also illustrate the use of counter-examples to show that a statement is false.

Click the link below to open Section 3 (17 pages, 374KB).

Section 3

4 Two identities

Section 4 introduces some important mathematical theorems.

Click the link below to open Section 4 (7 pages, 237KB).

Section 4

5 Solutions to the exercises

Section 5 contains solutions to the exercises that appear throughout sections 1-4.

Click the link below to open the solutions (13 pages, 500KB).

Section 5

Conclusion

This free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop
your approach to study and learning at a distance and helped to improve your confidence as an independent learner.

Keep on learning

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Acknowledgements

All other material contained within this course originated at the Open University.

Course image: craft_uas in Flickr made available under Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence.

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