Wk2 Notes PDF

ecture 3 og 124 A binaryoperationon a set A is a rule associating element y A foreachpair ixing A A is afunction Axa A ex E atbil...

ecture 3 og 124 A binaryoperationon a set A is a rule associating element y A foreachpair ixing A A is afunction Axa A ex E atbila.be catbi cadi cats cotas e e Need to checkthis so we need cates R obtas R since abind IR and IRisnosed underaddition thisistrue Ex A I a binder 8 List In definedtoearn since 8 A thisis a binaryoperation Ex A HIT H T flipa coin A J TT A binaryoperation ca O is associative itandonlyif a co c ca b c for an abic c a is commutative itandonlyif a b a abeA has a neutral eament ifandonly if there exists an element e e a sunthat a e a e a a taea we'd an sun an even the identity element 0 an element a ea has a 1 left inverse it andonly if there exists bea suchthat b a e sumeCA has identity carde 2 right inverse it andonly if there exists ceasuchthat a c e Q canyouhave at A such e a fae a a faff 1 a I is a setwithbinary operation t e Theorem It a a a ta c a the identityelementis unique if it exists proof Suppose e tea are identityelements for Thismeans a e a a and axte ava.ca andlinewise a e a a a a Kaea By in e f f and ca exit e butsince is a binary operation ext e so t e and the identity is unique a Theorem Let CA I be anassociative binary operation with the identity e Let a A If b is the leftinverse of a and c is the rightinverseof a then b c proof since is associative cy z ix y a x.y.ae A In particular can b e c guiseight fasiji I ceisthe identity so b c Definition of a group A group is a set 6 witha binary operation satisfying the following properties groupaxioms 61 is associative 62 thereexists e.co suchthat aone a e a a taco identityelement 63 foreveryenment a es sale osuchthat a a e and a a everyelementhas an inverse warning at Ya Ex CI.tl integerswithaddition eo a a Group ex Qt rational withaddition e o at a Group ex CIR I rearswithaddition e o a a Group  ex C I integerswith subtraction a as3 a i cas 2 notassociative noGroupcanson exC integers withmultiplication at II unless as nocroup can03 ex Q rational with multiplication doesnothaveaninverse o a noGroup tails03 wouldbe a it a Q1 a o p group Lecture 4 9125 Ex Groups 622 IR I A MatzUR I detCAIto 1 11 11 claim GLcars is agroup proof Firstnotethat is a binary operation ig ah 6LaCR Clearly since abioid IR thisproduct is a 2 2 IRmatrix and note that aCA B detcal deters to as AIB are invertible cat andtherefore A B isinvertible A Be612IR meaning is a binaryoperation Then we must mean a I II E II For 62 we e 98 i then 9 e Definition A group is abelion itthe binary operation is commutative Agroup is nonaberion it does not commute Theorem The identity enment in a group is unique Theorem Inversen in a group are unique leftinverse equalsthe rightinverse If aand b are inversen for g then a b proof consider that since o is group or implies carg b a garb in Moreover as areinveronog g ca g e and gabs e so as becomes e b a e andsince e isthe identity element b