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MethoδosPrimers,Vol. 1 TheaimoftheMethoδosPrimersseriesistomakeavailable conciseintroductionstotopicsin Methodology,Evaluation,Psychometrics,Statistics,DataAnalysis atanaffordableprice. Eachvolumeiswrittenbyexpertsinthefield, guaranteeingahighscientificstandard. MethoδosPublishers(UK) MethoδosVerlag(D) Sets, Relations, Functions IvoDüntsch SchoolofInformationandSoftwareEngineering UniversityofUlster Newtownabbey,BT370QB,N.Ireland [email protected] GüntherGediga FBPsychologie/Methodenlehre UniversitätOsnabrück 49069Osnabrück,Germany [email protected] Firstpublishedin2000 byMethoδosPublishers(UK), 24SouthwellRoad Bangor,BT203AQ c 2000byIvoDüntschandGüntherGediga. ° ISBN1903280001 ACIPrecordforthisbookisavailablefromtheBritishLibrary. IvoDüntschandGüntherGediga’srighttobeidentifiedastheauthorsofthiswork hasbeenassertedinaccordancewiththeCopyrightDesignandPatentsAct1988. All rights reserved. No part of this publication may be reproduced or transmitted in any form and by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writingfromthepublisher. Everyefforthasbeenmadetotracecopyrightholdersandobtainpermission. Any omissionsbroughttoourattentionwillberemediedinfutureeditions. TypesetinTimesNewRoman10pt. Manufacturedfromcamera-readycopysuppliedbytheauthorsby SächsischesDigitaldruckZentrum TharandterStraße31-33 D-01159Dresden Germany MethoδosPublishers(UK),Bangor MethoδosVerlag(D),Bissendorf Contents 1 Sets 7 1.1 IntroductiontoSets . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Subsets,PowerSets,EqualityofSets . . . . . . . . . . . . . . . . 10 1.3 FiniteandInfiniteSets . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 SetOperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 DeMorganRules,Distributivity,Tables . . . . . . . . . . . . . . 17 2 Relations 21 2.1 OrderedPairs,CartesianProducts . . . . . . . . . . . . . . . . . 21 2.2 IntroductiontoRelations . . . . . . . . . . . . . . . . . . . . . . 24 2.3 OrderingRelations . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 EquivalenceRelations . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Functions 35 3.1 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 One–one,Onto,andBijectiveFunctions . . . . . . . . . . . . . . 39 3.3 InverseFunctionsandPermutations . . . . . . . . . . . . . . . . 42 A SolutionstoExercises 45 Index51 Chapter 1 Sets 1.1 Introduction to Sets Mathematicsdealswithobjectsofverydifferentkinds;fromyourpreviousexperi- ence,youarefamiliarwithmanyofthem: Numbers,points,lines,planes,triangles, circles, angles, equations, functions and many more. Often, objects of a similar natureorwithacommonpropertyarecollectedintosets;thesemaybefiniteorin- finite(Forthemoment,itisenoughifyouhaveanintuitiveunderstandingoffinite, resp. infinite;amorerigorousdefinitionwillbegivenatalaterstage). Theobjects which are collected in a set are called theelements of that set. If an object a is an elementofasetM,wewrite a M ∈ whichisreadasa(isan)elementofM. IfaisnotanelementofM,thenwewrite a M 6∈ whichisreadasaisnotanelementofM. Example1.1.1. 1. IfQisthesetofallquadrangles,andAisaparallelogram, thenA Q. IfCisacircle,thenC Q. ∈ 6∈ 2. IfGisthesetofallevennumbers,then16 G,and3 G. ∈ 6∈ 3. IfListhesetofallsolutionsoftheequationx2 = 1,then1isanelementof L,while2isnot. 8 SETS, RELATIONS, FUNCTIONS Generally,therearetwowaystodescribeaset: Bylistingitselementsbetweencurlybracketsandseparatingthembycom- • mas,e.g. 0 , { } 2,67,9 , { } x,y,z . { } Notethatthisisconvenient(orindeedpossible)onlyforsetswithrelatively fewelements. Iftherearemoreelementsandonewantstolisttheelements “explicitly”sometimesperiodsareused;forexample, 0,1,2,3,... , 2,4,6,...,20 . { } { } Themeaningshouldbeclearfromthecontext. Inthisdescriptiveorexplicit method, an element may be listed more than once, and the order in which the elements appear is irrelevant. Thus, the following all describe the same set: 1,2,3 , 2,3,1 , 1,1,3,2,3 . { } { } { } By giving a rule which determines if a given object is in the set or not; this • isalsocalledimplicitdescription;forexample, 1. x : xisanaturalnumber { } 2. x : xisanaturalnumberandx > 0 { } 3. y : y solves(y+1) (y 3) = 0 { · − } 4. p : pisanevenprimenumber . { } Usually,thereismorethanonewayofdescribingaset. Thus,wecouldhave written 1. 0,1,2,... { } 2. 1,2,3,... { } 3. 1,3 {− } 4. 2 or x : 2x = 4 { } { } I. DÜNTSCH & G. GEDIGA 9 Thegeneralsituationcanbedescribedasfollows: Asetisdeterminedbyadefining propertyP ofitselements,writtenas x : P(x) { } whereP(x)meansthatxhasthepropertydescribedbyP. Theletterxservesasa variableforobjects;anyotherletter,orsymbolexceptP,wouldhavedoneequally well; similarly, P is a variable for properties or, as they are sometimes called, predicates. To avoid running into logical difficulties we shall always assume that our objects which are described by the predicate P come from a previously well defined set, say M, and sometimes we shall say so explicitly. In general then, we describesetsby x : x M andP(x) , { ∈ } whichalsocanbewrittenas x M : P(x) . { ∈ } Weshallusethefollowingconventionsindescribingcertainsetsofnumbers: N = 0,1,2,3,... isthesetofnaturalnumbers. • { } N+ = 1,2,3,... isthesetofpositivenaturalnumbers. • { } Z = ..., 3, 2, 1,0,1,2,3,... isthesetofintegers. • { − − − } Q = x : x = a , where a Z,b Z, and b = 0, is the set of ratio- • { b} ∈ ∈ 6 nal numbers; observe that each rational number is the ratio of two integers, whencethename. R = x : xisarealnumber . • { } Weshallnotgivearigorousdefinitionofarealnumber;itisassumedthatyouhave anintuitiveideaofthereals-thinkofthemasbeingthepointsonastraightline. 1.1.1 Exercises Exercise1. Describethefollowingsetsexplicitly: A = x Z : x (cid:12) 5 { ∈ } B = x N : xdivides24 { ∈ } C = x R : x = 2 { ∈ } + D = x N : x (cid:12) 0 { ∈ }