# Mathematics Hl Option Sets Relations And Groups Pdf

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No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available ISBN 10 9 8 7 6 5 4 3 2 1 Printed in Great Britain by Bell and Bain Ltd, Glasgow Paper used in the production of this book is a natural, recyclable product made from wood grown in sustainable forests.

The manufacturing process conforms to the environmental regulations of the country of origin. They will help students CAS. They individuals and communities. They are brave and articulate in decisions. The another student. Examples include, essay. Did you know? History Mathematics education is a growing, ever changing entity.

During this time, the past 1 6 years at Bonn International School. She joined the IB the IB programme since 1 In his conjecture Cantor says that there is no set whose size is between 0 and 1. In some books it is denoted by card S or S. One method to show that two sets A and B are equal is called the containment method, or the double inclusion method. To show that two sets A and B are equal we need to show both containment conditions, i.

Since m is an integer, k is also an integer. Write down the statement. Since we proved that P0 is true and we showed that Write fnal statement.

Who shaves the barber? This paradox arises because Russell tries to fnd the set containing all sets. On the right, the seashells have been organized by type.

The seashells have been partitioned into sets which are disjoint but together make up the whole set. Let A be a non-empty set. The sets represent all the continents and each country belongs to one continent only. The sets in a partition must be disjoint.

Armenia, for example, is in both set A and set B. The given sets do not partition S because the The sets in a partition must be disjoint. Exercise 1B 1 A deck of playing cards contain 52 cards. These are divided into two red suits hearts and diamonds and two black suits spades and clubs. Each suit contains 13 cards representing the numbers 1 to 10 plus three picture cards Jack, Queen and King. The picture on the next page shows a deck of cards partitioned into 4 suits. List a further two ways in which you could partition a deck of cards.

This will help you visualize what you are aiming to prove. The proof is left as an exercise. Set properties Before we move on, we need to prove some properties of sets that will be used in the rest of the book. The following theorem concerns properties that may seem trivial. These basic properties will be required for proofs of less obvious results.

Part ii is included in the next exercise. Sometimes it is easier to prove a statement by using set properties than by using the double inclusion method. The previous theorems are essential when proving complex results, especially when the double inclusion method becomes too cumbersome. This is illustrated in the next example. You may frst want to draw Venn diagrams to help visualize what you are trying to prove. Hint: Use the double inclusion method used in Example 4.

Looking at a corner in the ceiling he saw three lines and three planes which intersected at the corner. Descartes had created a system to describe 3D space. Now that you understand what a Cartesian product is we can move on to appreciate how this product allows us to construct other sets.

Are the two products equal? List all the elements that make up the relation R. The relation is said to be symmetric. The arrows indicate the relation B. Is R an equivalence relation? There are other examples that you might be able to come up with. Symmetric because 1 R2 and 2R1 and 2R3 and 3R2. Not symmetric because 1 R2 but 2 R 1. Show that R is an equivalence relation. Any two numbers in the same column are congruent to each other modulo 6. Although the table represents only the integers 1 to 60, it is clear that we could continue to build up the table endlessly.

All the positive integers could be included in such an endless table, and they would all be separated into distinct equivalence classes representing the particular congruence.

Example 17 For each given set S and associated relation R, determine whether or not R is an equivalence relation. R is an equivalence relation. R is refexive. R is symmetric. R is not transitive. So R is symmetric.

R is not refexive. We could also have chosen any other non-zero natural number for x. Show that R is an equivalence relation on S. Show that R is symmetric but not refexive or transitive. Determine whether or not R is an equivalence relation. We can also illustrate this by sketching a diagram. It is easy to check that R is refexive, symmetric and transitive. So R is an equivalence relation on S. We can illustrate the relation on a diagram.

Check the properties for an equivalence relation. Assume that two equivalence classes [xi ] and [xj ] are not disjoint. Now we need to prove that the equivalence classes are exhaustive, The most trivial i. This means that the equivalence classes [xi ] partition the set A.

In a and b, show that R is an equivalence relation and list the equivalence classes induced by each relation on W. Show that this is an equivalence relation on L and describe the partition induced by R.

Show that R is an equivalence relation and describe the partitions induced by R. Show that R is an equivalence relation and describe the partition induced by R on P.

Describe the equivalence class [ l, 2 ]. Describe the equivalence class [ l, l ]. Hence or otherwise describe the partition induced by R. Use Venn diagrams to illustrate the distributive laws. Determine whether or not S is transitive.

AB is a directed line segment where A is the starting point and B is the terminal point. Equivalence classes are mutually exclusive and the set A is partitioned into equivalence classes by an equivalence relation R on A.

Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse. Proofs of the uniqueness of the identity and inverse elements. He used it to describe quantities related to curves. We can represent this pictorially as shown here. The set S is called the domain and T, the target set, is called the co-domain. Proved by contradiction. They make sure that there are no contradictory or ambiguous connotations.

In other words the frst rule ensures that we do not have any singularities. In other in the co-domain. The two examples below illustrate surjections.

## Mathematics Higher Level for the IB Diploma Option Topic 8 Sets, Relations and Groups

Prints of this textbook before included a student CD, but any later reprints onwards have moved all of the CD features to Snowflake - our online learning system. Together, they aim to provide students and teachers with appropriate coverage of the two-year Mathematics HL Course, first examined in The aim of this topic is to introduce students to the basic concepts, techniques and main results in abstract algebra, specifically for sets, relations and group theory. Detailed explanations and key facts are highlighted throughout the text. Each sub-topic contains numerous Worked Examples, highlighting each step necessary to reach the answer for that example. In this changing world of mathematics education, we believe that the contextual approach shown in this book, with associated use of technology, will enhance the student's understanding, knowledge and appreciation of mathematics and its universal applications.

This is for high-achiever. Questions are sometimes harder than IB past papers. House construction in japan. These syllabuses enable learners to extend the mathematics skills, knowledge, and understanding developed in the Cambridge IGCSE or O Level Mathematics courses, and use skills in Posted by. Worked Solutions Math HL.

Mathematics Higher Level for the IB Diploma Option Topic 8 Sets, Relations and Groups. Pages · · KB ·.

## Mathematics Higher Level for the IB Diploma Option Topic 8 Sets, Relations and Groups

Past Papers. Nov Examination Schedule. This subreddit is for all things concerning the International Baccalaureate, an academic credential accorded to secondary students from around the world after two vigorous years of study, culminating in challenging exams. This subreddit encourages questions, constructive feedback, and the sharing of knowledge and resources among IB students, alumni, and teachers.

Hons , Grad. Cover design by Piotr Poturaj. Typeset in Australia by Deanne Gallasch.

It is also for use with the further mathematics course. Based on the new group 5 aims, the progressive approach encourages cumulative learning. Features include: a dedicated chapter exclusively for mixed examination practice- plenty of worked examples- questions colour-coded according to grade- exam-style questions- feature boxes throughout of exam hints and tips. Mind you, that doesn't mean the relation was poorly written; it's once you've gotten it that everything makes sense in an incredible way. Look at your relation, it's leather, it's like Groups man, your skin.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization.

### Mathematics HL (Option): Sets, Relations and Groups

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4 FOREWORD Mathematics HL (Option): Sets, Relations and Groups has been The accompanying student CD includes a PDF of the full text and access to.

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Она понимала, что это больше не имеет значения: Хейл и без того знал все, что можно было знать. Мне нужно доложить об этом Стратмору, - подумала она, - и как можно скорее. ГЛАВА 38 Хейл остановился в центре комнаты и пристально посмотрел на Сьюзан. - Что случилось, Сью. У тебя ужасный вид.