Abstract Algebra MATH 3330
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This 11 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 3330 at University of Houston taught by Staff in Fall. Since its upload, it has received 109 views. For similar materials see /class/208414/math-3330-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
A78 6 Math 3330 Section 14 Binary Operations A binary operation on a nonempty set A is a mapping f from A X A into A Examples 0amp1 JO 3 Q Traditional Integers With addition Ck g amp b Real Number With multiplication CL I b e 9 X K 5 DL 4 E NonTraditional Examples The set of mappings f I Z gt Z With function composition 6 73 6 WM KM o gt g g The power set of a nonempty set A With set intersection 909 m 33 7 m Notation for binary operations Ck Eb Properties of a system consisting of a set With one or more binary operations 7 note that these properties depend both on the operation AND the set A A vthfx yfbcy erlt NVLiXlkig ll X Avg gtlt 7ampampgzgt Commutativity Associativity Closure of the binary operation with respect to a subset 5 Q A A S LKoSeJ cu 1C 7 5 4 YK7 6 g X f L S Raw 2 3 CW1 JV woth C O Q Inrgwg 14274 Identity Element 0Q KL Vt g 07 Were BVYIS 8 mot Mm vx 69 7C r E r 8 W X 7C Ems wmw 0 WW Right Inverse A 6 a W X AOL5 A W WW wAVWSQ LeftInverse 7C Aqg Cr EgtL MVVSe amp I 9 Z 4 X r 6 Inverse of an element I t g 3 Cf I A wwmw x4 Mc y A X w 695 Cage K 5 CM IVQ IL C M 7 lg X has AW 1 NW Mme9Q Chi K 0k E averse t ef 410 Tlt SW IOKOCI gt5 r e a 3 x a W Cy g 1 29m w5tz tx amp 6 Notation The set A With the binary operation will be denoted A Examples Determine which properties the following systems possess lt1 X Win K Yes Assocha w b 3 65 dose Q 1 YES KW k 7 N0 9 Irwgr zs 51 M M MW e l g1gtltf a nr NJQH Wenvb Lw s wkKW om Z Wherexyxy3 forallintegersxandy CommaK 39ma W25 X 7 w X 7 3 1 gtlt 1L 3 A oc cdlvc 85 k57kg xx91 37 3 I X Jet Z gixyw X yw x A g g 3 7 gg3gt3wlt97c9 W9 lJE becaues Yin3 GE rsfAc C om EMU7L Ma gtlt A e Z X Xe 43 06 S 8 63 Ime3935 gtlt CLT SMJ7LW X 19 g e 3 3 1 3 ex VCR y X 9 Y k xe x we w mj V d RWhereXyIX yforallrealnumbersxandy W iwit y x X Kg 2 9 x ssod A 3 vaira 3r0 g CFquot Z1 3 fr 1 50 31 4 97 33to Cosaamp bqgcukgg Klt 6 R NE f 5 e gtltf r Jf C lf amp K MQMF Fr me 700 I t T e W 7 7333 ie ffi zwherexy2xy WW 5 VW 25 HQ ya ykwt WygtO sze Assocr oUCVVE myng Lexwv Hz Xkykagx4y4g absd QR is am We Km UO E 7H L i r 23 gt Has mo 30 P W for a nonempty set Ownrm fitBfg sgocmm CgtEG41C Cage l A 5 KW ang 85 gag B NC INKage G C MT S S Cnw 5 63 ya 399 90W 75 Assume that is an associative binary operation on A With an identity element Prove that the inverse of an element is unique When it exists 8 A 5 f mxs e qA deEDthgk e inverse was Xt gtlt 5W E XX ze M ycg giretat w ggtlt3z 2 Math 3330 Section 12 Mappings An ordered pair ab ab cd ifand only if ac and Ed The Cartesian Product of sets A and B Vd yu SLEYi AgtltBabaeAbeB s a 07050 Aj W is NOT WIW asa Axg gcslammigiw Let A and B be nonempty sets A subset f of A X B is a mapping from A to B ifand onlyiffor each a E A there is aunique b E B suchthat a b E f We also Write b fa and call b the image ofa under f mam The set A is called the domain off and B is called the codomain off The image off Written fA is the set CyeByfa forsomeaeA A 1356 and B 27 Subsets and Mappings Let S g A and let f I A gt B be amapping Then fS y e B y fs for some s e S is calledthe image of S under f Let T g B Then f 1T x e A fx 6 T The SET f 1T is called the inverse image ofT under f Composition of Mappings Just as happens in function notation in calculus mappings can be put together by composition For g I A gt B f I B gt C the composition fog A gt C is de nedby f o ga fgafor all a e Aw a Lq b 69 Special Mappings Gk 6441f Onto 7 Surjection Mappings A map f I A gt B is called surjective or onto if and only iffAB 6quot5 l G Onetoone 7 lnjective Mappings A map f I A gt B is called injective or onetoone if and only ifdifferent elements of A always have different images under f Bijective Mappings 7 Onetoone correspondence Amap f I A gt B is called bijective ifand only ifit is surjective and injective Another name for a bijective map is a onetoone correspondence between A and B Picture Representations of Mappings 5 ii 5 on i o Md M39i39om 1b quot i mCa lBls A 3 06 Ox MWW rm wm anges 77421345 Lquot 9 f a x JD V 53 ce Mappings are most often represented by a formula for the image of an element of A This is equivalent to functions you should be very familiar with from previous math classes The new twists will appear in later chapters when we require our maps to have other special properties Let s practice working with sets and maps Notice that the de nitions for injective and surjective maps depend on the formula for the mapping AND the sets involved We are NOT assuming that our functions are given for all real numbers only for the sets of numbers speci ed E am ples Determine if the following maps are surjective injective or bijective Q r E 5 E Leg KM v 6 1 Y lif TW 1 v o 7 1 7 bljcdllv 1 15 lnje c l e lvlo q W I t X X 4X xgl X2l 3957 g 6 gt K 6 X7 5 W W k lgtlt V C R 7 I goat X QX1AIZ Qxlhi axltayz M2 1 g quota 70 axq MAM 7xv tf 2Xh QXS gtlttS fnj adrch 39 Par QCLJL fZ gtZfx E xeve 2x xodd 5 onivo beccuusc g g 502Z amp 97 5 7 1 6m 93 we Mfg W z gm g 40 E W 27LfoJOW 7Lfvvi ga11 1E 3 E wm fo Xamp ng3w X 7 6 gt6 Mk vgm W XZXgeg gwnc m 912 a xx2 E gz AEandBE Egt E g I 5 H m fo m is NOT WWgt 2 c 21X X WMX 6 X9 W M KC39E M M gtltgtgt2 4 ogtlt4 3 XO X 737 2X3 is x LetfZ gtZbegivenbyfx X 1 x ven 2X xodd Alagt17 erg 7 X s 0 1 2 nd fS f 1fs 6 Jah gCS 3433 Ojia 67V9gt50252 6amp0 9amp4 3 308 T 5 a nd f 1T ff 1T 6 M M z o QX2Q gt3 T c f3 quot gt 095 56gt Let f I A gt B With A and B nonempty Prove that f has the property that f71f S for every subset S ofA ifand only iffis onetoone AW w s vg kw gtltU5lt97 2 Clst 6Kg023 M 55x 6quot 57gtgt3 3m gm 3 gay 53 Stage 60 6 57 X36 AX M C l 7 V EV j XQEX 3 m39tohm f ASSW aka 7ij jmw g hg fgg AX s 34 A 39ff4 TMamp 5 g s r m L 4W 3 v39 6 gtgt 35 mug 0 Wquot gsgtgts Qmmm w eg wwsp Lg amp g 563 5WAMt mw 66x gt imam 4 K Mg 73me x T XEJS
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