In this final section we look at a method of classifying the elements of a set by sorting them into subsets. We shall require that the set is sorted into disjoint subsets â€“ so each element of the set belongs to exactly one subset. Such a classification is known as a partition of a set. In order to achieve a partition, we need to have a method which enables us to decide whether or not one element belongs to the same subset as another. We look first at the general idea of a r
Author(s): The Open University

3.4 Modular multiplication

In the last subsection we stated that, for any integer n â‰¥ 2, the set n satisfies the same rules for addition modulo n as the real numbers satisfy for ordinary addition. When it comes to multiplicat
Author(s): The Open University

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.

Author(s): The Open University

3.5 Scaling areas and volumes

In OpenLearn course Diagrams, graphs and charts you saw how a scale is used on plans of houses and other structures. The scale makes it possible to take a length on the plan and calculate the correspond
Author(s): The Open University

Try some yourself

## Question 1

Which of these triangles are similar?

2.9 Similar and congruent shapes

Two shapes are said to be similar if they are the same shape but not necessarily the same size. In other words, one may be an enlargement of the other. They may also have different orientations, as in the drawing below.

4.1 Introduction

We have seen in SAQ 18 of Section 3.4 how some sets of points of the complex plane can be described algebraically in terms of operations on C. We now use the modulus function to take this a step further by defining discs in the complex plane. As we shall see, discs are extensively used in arguments involving l
Author(s): The Open University

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
Author(s): The Open University

A Non-Libertarian Approach to Drug Legalization

Hosted by Young Americans for Liberty at Auburn University and the Mises Institute. Recorded 20 April 2019 at the Mises Institute in Auburn, Alabama.

Author(s): No creator set

World in transition: Migration and Trade
What does it mean to be poor, or an immigrant? What form should Aid take? This album begins to explore the complex issues of international development in a globalised world, starting with a look at schemes which attempt to alleviate poverty. Small business owners are empowered by micro-financing in Glasgow, while in Argentina a disastrous economic collapse has led to people taking matters into their own hands by creating an alternative social exchange currency. Migrant communities, such as Greek
Author(s): The OpenLearn team

Introduction

Access to safe, clean and sustainable energy supplies is one of the greatest challenges facing humanity during the twenty-first century. This course will survey the world's present energy systems and their sustainability problems, together with some of the possible solutions to those problems and how these might emerge in practice.

Throughout history, the use of energy has been central to the functioning and development of human societies. But during the nineteenth and twentieth centuri
Author(s): The Open University

Introduction

Nature Matters considers environmental responsibility and what may matter from a caring perspective and an accountability perspective. A reading by Andrew Light reflects on four key debates in environmental ethics regarding the way in which nature is valued, and prompts the question on how such debates might inform environmental responsibility.

Section 2 examines the formal processes involved in developing accountability in the context of sustainable development. The persuasiveness of t
Author(s): The Open University

Sustainable Scotland
Sustainable Scotland is a free course that will appeal to anyone with an interest in a sustainable future in the context of contemporary Scottish society. It will give you a broad-based introduction to a number of different aspects of sustainability that impact on Scotland and the wider world. First published on Wed, 30 Mar 2016 a
Author(s): Creator not set

Introduction

This course provides an introduction to global warming. We will be considering the history of global warming by looking at the pattern of ice ages and analysis of recorded temperatures. We will aim to gather meaningful information from this data. We will briefly assess the impact and influence of humans on global warming and, finally, we will examine climate models and how to predict future changes.

This OpenLearn course provides a sample of postgraduate study in Author(s): The Open University

Organisations, environmental management and innovation
Do you want to discover more about how organisations innovate for environmental management? This free course, Organisations, environmental management and innovation, focuses on the innovations organisations are developing or using to manage contemporary environmental issues. It also covers how organisations are innovating thinking about environmental management. Author(s): Creator not set

Earth's physical resources: petroleum
The discovery of the world's first major underground oil field in Pennsylvania, USA in 1859 sparked the continuing era of the world's reliance on cheap energy from oil and gas. This free course, Earth's physical resources: petroleum, begins by examining the geological characteristics of petroleum and the key ingredients necessary to form oil and gas accumulations. Then there is a brief description of industrial operations during the life cycle of an oil field, starting with subsurface analysis a
Author(s): Creator not set

Keep on learning

## Study another free course

There are more than 800 courses on OpenLearn for you to choose from
Author(s): The Open University

2.7 Summary

1. Reconstructions based on direct temperature measurements (back to 1860) and proxy data (Box 7) reveal that the Earth's GMST varies naturally on many different time-scales: from year-to-year, over periods of several decades and, in the longer term, according to the roughly 100 000-year rhythm of glacial/interglacial cycles. The past 10 000 years has been marked by the relatively stable global climate of the present interglacial (the Holocene).

2. Author(s): The Open University

2.3 Contested science: a case study

For complex issues such as global climate change, there are many opportunities for scientists to take issue with the findings of their colleagues. They can disagree about the procedures for gathering data, the completeness or coverage of the data, how the data are analysed and interpreted, and then finally the conclusions. The assumptions that shape a particular piece of research and inform the kind of questions that will be asked can be no less contentious than the quality of the data gather
Author(s): The Open University

2.2 Records of the Earth's temperature

To put the temperature records reported by the IPCC in context, we start with a longer-term geological perspective on the Earth's GMST.

Author(s): The Open University