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Introduction

In this unit we use the geometric concept of symmetry to introduce some of the basic ideas of group theory, including group tables, and the four properties, or axioms, that define a group.

Please note that this unit is presented through a series of PDF documents.

This unit is an adapted extract from the Open Unviersity course Pure ma
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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).

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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 ex
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4.2 Defining useful subsets of the complex number system, and proving the Nested Rectangles Theorem

You will no doubt recall that in real analysis extensive use is made of the modulus function . It gives us a way of measuring the “closeness” of two numbers, which we exploit in writing expressio
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7 Solutions to the exercises

Section 6 contains solutions to the exercises that appear throughout sections 1-5.

Click the link below to open the solutions (13 pages, 232KB).

Section 6


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Learning outcomes

After studying this course, you should be able to:

  • understand the definition of a real function

  • use the notation for intervals of the real line

  • recognise and use the graphs of the basic functions described in the audio section

  • understand the effect on a graph of translations, scalings, rotations and reflections

  • understand how the shape of a graph of a function features properties of the function such as increasing, decr
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3.1.1 Try some yourself

1 Express each of the following percentages as fractions:

  • (a) 40%

  • (b) 8%

  • (c) 70%

  • (d)
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2.2 Direct proportion

In a recipe the quantity of each ingredient needed depends upon the number of portions. As the number of portions increases, the quantity required increases. The quantity per portion is the same. This is called direct proportion. The quantity is said to be directly proportional to the number of portions. If 2 potatoes are required for one portion, 4 will be required for two portions etc. A useful method for direct proportion problems is to find the quantity for one and multiply by the
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3.2.1 Try some yourself

Activity 20

Convert each of the following to percentages. Round off the percentages to whole numbers.

  • (a)

    • (i) 0.8

    • (ii) 0.
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1.3 Square roots

Given any number, you now know how to find its square. But, given the squared number, how do you find the original number?

Example 3

Learning outcomes

After studying this course, you should be able to:

  • understand and use the basic terms for the description of the motion of particles: position, velocity and acceleration

  • understand, use and differentiate vector functions

  • understand the fundamental laws of Newtonian mechanics

  • solve mechanics problems in one dimension by drawing a sketch, choosing a suitable x-axis and origin, drawing a force diagram, applying Newton’s second law, tak
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Learning outcomes

After studying this course, you should be able to:

  • appreciate the concept of force, and understand and model forces such as weight, tension and friction

  • model objects as particles or as rigid bodies, and the forces that act on an object in equilibrium

  • use model strings, rods, pulleys and pivots in modelling systems involving forces

  • understand and use torques

  • model and solve a variety of problems involving systems in equ
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1.5 Exercises

Exercise 1

A vector a has magnitude |a| = 7 and direction θ = −70°. Calculate the component form of a, giving the components correct to two decimal places.

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1.1: Converting to component form

In some applications of vectors there is a need to move backwards and forwards between geometric form and component form; we deal here with how to achieve this.

To start with, we recall definitions of cosine and sine. If P is a point on the unit circle, and the line segment OP makes an angle θ measured anticlockwise from the positive x-axis, then cos θ is the x-coordinate of P and sin θ is the y-coordinate of P (
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Learning outcomes

After studying this course, you should be able to:

  • convert a vector from geometric form (in terms of magnitude and direction) to component form

  • convert a vector from component form to geometric form

  • understand the use of bearings to describe direction

  • understand the difference between velocity and speed

  • find resultant displacements and velocities in geometric form, via the use of components.


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Acknowledgements

All materials included in this course are derived from content originated at the Open University.

Course image: fdecomite in Flickr made available under Creative Commons Attribution 2.0 Licence.

Don't miss out:

If reading this text has inspired you to learn m
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1.5 Studying the Möbius band

Task 10 The Möbius band

Take a long thin strip of paper (preferably squared or graph paper) about 30 cm by 3 cm. Give one end a half twist and then tape it together. This is a Möbius band as shown in Author(s): The Open University

1.4.4 O is for Objectivity

One of the characteristics of ‘good’ information is that it should be balanced and present both sides of an argument or issue. This way the reader is left to weigh up the evidence and make a decision. In reality, we recognise that no information is truly objective.

This means that the onus is on you, the reader, to develop a critical awareness of the positions represented in what you read, and to take account of this when you interpret the information. In some cases, authors may
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1.3.1 Introduction

You can find a lot of information about the maths and statistics on the internet.

To find this information you might choose to use:

  • search engines and subject gateways;

  • books and electronic books;

  • databases;

  • journals;

  • encyclopedias

  • internet resources


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Studying mammals: The social climbers
Monkeys have long fascinated us because of their similarities to the human race. In this free course, Studying mammals: The social climbers, you will find out about some of the characteristics that make them so like us: their physiology, complex social interactions, large brains and intelligence. This is the ninth course in the Studying mammals series.Author(s): Creator not set

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Except for third party materials and otherwise stated (see http://www.open.ac.uk/conditions terms and conditions), this content is made available under a http://creativecommons.org/licenses/by-nc-sa/2

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