RSS feed for Mathematical language
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection0
This RSS feed contains all the sections in Mathematical language
Moodle
Copyright © 2016 The Open University
https://www.open.edu/openlearn/ocw/theme/image.php/_s/openlearnng/core/1563274092/i/rsssitelogo
moodle
https://www.open.edu/openlearn/ocw
140
35
engbWed, 17 Jul 2019 15:00:45 +0100Wed, 17 Jul 2019 15:00:45 +010020190717T15:00:45+01:00The Open UniversityengbCopyright © 2016 The Open UniversityCopyright © 2016 The Open University
Introduction
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection0
Tue, 12 Apr 2016 23:00:00 GMT
<p>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.</p><p>In mathematics we try to avoid these difficulties by expressing our thoughts in terms of welldefined 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.</p><p>Please note that this course is presented through a series of downloadable PDF files.</p><p>This OpenLearn course is an adapted extract from the Open Unviersity course <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www3.open.ac.uk/study/undergraduate/course/m208.htm?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">M208: <i>Pure Mathematics</i></a></span></p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection0
IntroductionM208_5<p>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.</p><p>In mathematics we try to avoid these difficulties by expressing our thoughts in terms of welldefined 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.</p><p>Please note that this course is presented through a series of downloadable PDF files.</p><p>This OpenLearn course is an adapted extract from the Open Unviersity course <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www3.open.ac.uk/study/undergraduate/course/m208.htm?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">M208: <i>Pure Mathematics</i></a></span></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University

Learning outcomes
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsectionlearningoutcomes
Tue, 12 Apr 2016 23:00:00 GMT
<p>After studying this course, you should be able to:</p><ul><li><p>use set notation</p></li><li><p>determine whether two given sets are equal and whether one given set is a subset of another</p></li><li><p>find the union, intersection and difference of two given sets</p></li><li><p>determine the image of a given function</p></li><li><p>determine whether a given function is oneone and/or onto.</p></li></ul>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsectionlearningoutcomes
Learning outcomesM208_5<p>After studying this course, you should be able to:</p><ul><li><p>use set notation</p></li><li><p>determine whether two given sets are equal and whether one given set is a subset of another</p></li><li><p>find the union, intersection and difference of two given sets</p></li><li><p>determine the image of a given function</p></li><li><p>determine whether a given function is oneone and/or onto.</p></li></ul>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University

1 Sets
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection1
Tue, 12 Apr 2016 23:00:00 GMT
<p>In Section 1 we discuss the idea of a <i>set</i> 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 <i>union</i>, <i>intersection</i> and <i>difference</i>.</p><p>Click the link below to open Section 1 (16 pages, 389KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26497">Section 1</a></span></p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection1
1 SetsM208_5<p>In Section 1 we discuss the idea of a <i>set</i> 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 <i>union</i>, <i>intersection</i> and <i>difference</i>.</p><p>Click the link below to open Section 1 (16 pages, 389KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26497">Section 1</a></span></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University

2 Functions
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection2
Tue, 12 Apr 2016 23:00:00 GMT
<p>In Section 2 we give the general definition of a <i>function</i>, 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 <i>composing</i> two functions, and the idea of forming the <i>inverse</i> of a function.</p><p>Click the link below to open Section 2 (16 pages, 366KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26498">Section 2</a></span></p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection2
2 FunctionsM208_5<p>In Section 2 we give the general definition of a <i>function</i>, 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 <i>composing</i> two functions, and the idea of forming the <i>inverse</i> of a function.</p><p>Click the link below to open Section 2 (16 pages, 366KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26498">Section 2</a></span></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University

3 The language of proof
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection3
Tue, 12 Apr 2016 23:00:00 GMT
<p>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 <i>direct proof</i>, <i>proof by mathematical induction</i>, <i>proof by contradiction</i> and <i>proof by contraposition</i>. We also illustrate the use of <i>counterexamples</i> to show that a statement is false.</p><p>Click the link below to open Section 3 (17 pages, 374KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26499">Section 3</a></span></p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection3
3 The language of proofM208_5<p>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 <i>direct proof</i>, <i>proof by mathematical induction</i>, <i>proof by contradiction</i> and <i>proof by contraposition</i>. We also illustrate the use of <i>counterexamples</i> to show that a statement is false.</p><p>Click the link below to open Section 3 (17 pages, 374KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26499">Section 3</a></span></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University

4 Two identities
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection4
Tue, 12 Apr 2016 23:00:00 GMT
<p>Section 4 introduces some important mathematical theorems.</p><p>Click the link below to open Section 4 (7 pages, 237KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26501">Section 4</a></span></p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection4
4 Two identitiesM208_5<p>Section 4 introduces some important mathematical theorems.</p><p>Click the link below to open Section 4 (7 pages, 237KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26501">Section 4</a></span></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University

5 Solutions to the exercises
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection5
Tue, 12 Apr 2016 23:00:00 GMT
<p>Section 5 contains solutions to the exercises that appear throughout sections 14.</p><p>Click the link below to open the solutions (13 pages, 500KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26500">Section 5</a></span></p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection5
5 Solutions to the exercisesM208_5<p>Section 5 contains solutions to the exercises that appear throughout sections 14.</p><p>Click the link below to open the solutions (13 pages, 500KB).</p><p><span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=26500">Section 5</a></span></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University

Conclusion
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection6
Tue, 12 Apr 2016 23:00:00 GMT
<p>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.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection6
ConclusionM208_5<p>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.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University

Keep on learning
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection7
Tue, 12 Apr 2016 23:00:00 GMT
<div class="oucontentfigure oucontentmediamini"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/89614/mod_oucontent/oucontent/745/8ff4c822/d3c986e6/ol_skeleton_keeponlearning_image.jpg" alt="" width="300" height="200" style="maxwidth:300px;" class="oucontentfigureimage"/></div><p> </p><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Study another free course</h2><p>There are more than <b>800 courses on OpenLearn</b> for you to choose from on a range of subjects. </p><p>Find out more about all our <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">free courses</a></span>.</p><p> </p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Take your studies further</h2><p>Find out more about studying with The Open University by <a class="oucontenthyperlink" href="http://www.open.ac.uk/courses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">visiting our online prospectus</a>. </p><p>If you are new to university study, you may be interested in our <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/doit/access?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Access Courses</a> or <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/certificateshe?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Certificates</a>.</p><p> </p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">What’s new from OpenLearn?</h2><p>
<a class="oucontenthyperlink" href="http://www.open.edu/openlearn/aboutopenlearn/subscribetheopenlearnnewsletter?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Sign up to our newsletter</a> or view a sample.</p><p> </p></div><div class="oucontentbox oucontentshollowbox2 oucontentsbox oucontentsnoheading "><div class="oucontentouterbox"><div class="oucontentinnerbox"><p>For reference, full URLs to pages listed above:</p><p>OpenLearn – <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a>
</p><p>Visiting our online prospectus – <a class="oucontenthyperlink" href="http://www.open.ac.uk/courses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses</a>
</p><p>Access Courses – <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/doit/access?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses/<span class="oucontenthidespace"> </span>doit/<span class="oucontenthidespace"> </span>access</a>
</p><p>Certificates – <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/certificateshe?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses/<span class="oucontenthidespace"> </span>certificateshe</a>
</p><p>Newsletter ­– <a class="oucontenthyperlink" href=" http://www.open.edu/openlearn/aboutopenlearn/subscribetheopenlearnnewsletter?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>aboutopenlearn/<span class="oucontenthidespace"> </span>subscribetheopenlearnnewsletter</a>
</p></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsection7
Keep on learningM208_5<div class="oucontentfigure oucontentmediamini"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/89614/mod_oucontent/oucontent/745/8ff4c822/d3c986e6/ol_skeleton_keeponlearning_image.jpg" alt="" width="300" height="200" style="maxwidth:300px;" class="oucontentfigureimage"/></div><p> </p><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Study another free course</h2><p>There are more than <b>800 courses on OpenLearn</b> for you to choose from on a range of subjects. </p><p>Find out more about all our <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">free courses</a></span>.</p><p> </p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Take your studies further</h2><p>Find out more about studying with The Open University by <a class="oucontenthyperlink" href="http://www.open.ac.uk/courses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">visiting our online prospectus</a>. </p><p>If you are new to university study, you may be interested in our <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/doit/access?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Access Courses</a> or <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/certificateshe?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Certificates</a>.</p><p> </p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">What’s new from OpenLearn?</h2><p>
<a class="oucontenthyperlink" href="http://www.open.edu/openlearn/aboutopenlearn/subscribetheopenlearnnewsletter?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Sign up to our newsletter</a> or view a sample.</p><p> </p></div><div class="oucontentbox oucontentshollowbox2 oucontentsbox
oucontentsnoheading
"><div class="oucontentouterbox"><div class="oucontentinnerbox"><p>For reference, full URLs to pages listed above:</p><p>OpenLearn – <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a>
</p><p>Visiting our online prospectus – <a class="oucontenthyperlink" href="http://www.open.ac.uk/courses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses</a>
</p><p>Access Courses – <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/doit/access?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses/<span class="oucontenthidespace"> </span>doit/<span class="oucontenthidespace"> </span>access</a>
</p><p>Certificates – <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/certificateshe?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses/<span class="oucontenthidespace"> </span>certificateshe</a>
</p><p>Newsletter – <a class="oucontenthyperlink" href=" http://www.open.edu/openlearn/aboutopenlearn/subscribetheopenlearnnewsletter?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>aboutopenlearn/<span class="oucontenthidespace"> </span>subscribetheopenlearnnewsletter</a>
</p></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University

Acknowledgements
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsectionacknowledgements
Tue, 12 Apr 2016 23:00:00 GMT
<p>All other material contained within this course originated at the Open University.</p><p>Course image: <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.flickr.com/photos/craft_uas/">craft_uas</a></span> in Flickr made available under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/byncsa/2.0/legalcode">Creative Commons AttributionNonCommercialShareAlike 2.0 Licence</a>.</p><p><b>Don't miss out:</b></p><p>If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University  <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a></p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsandstatistics/mathematics/mathematicallanguage/contentsectionacknowledgements
AcknowledgementsM208_5<p>All other material contained within this course originated at the Open University.</p><p>Course image: <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.flickr.com/photos/craft_uas/">craft_uas</a></span> in Flickr made available under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/byncsa/2.0/legalcode">Creative Commons AttributionNonCommercialShareAlike 2.0 Licence</a>.</p><p><b>Don't miss out:</b></p><p>If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University  <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBMathematical language  M208_5Copyright © 2016 The Open University