Math 3306 - Test 1 Name: An d {"0v\ ( _ roj ~ ed Date: l'( ~0 { 1\ Fall 2011 1. (to Pts) Let S == {I, 2, 3,4, 5,6,7,8,9, 10}. of each of the following types of mappings, provide justification for why the example satisfies the conditions. // 1 - --::: -.J r l/... a. Give an example a bijection a : S ~ s. ~{..\ ~ \Ji " (..' 1)') I '" ):( 0' Vl1r V\ ->\('~ j J) -:.' d' _ -:., 11 <):' / b. Describe your above bijection as a permutation of th ments of S in 2-row notation and also in cyclic notation.,-. ( I. 2. S '-1- S <:, '1 '3 "1 \ ~ '2- 'S '-t S ' I ~ 0 ) ( I) Cl c,\. c. 1\'0\ ~ '~O'-_ **** The 2 theorems below will be used for reference **** Theorem 2.1 Assume that ex: S ~ T and {3( T ~ U. (a) If ex and (3 are onto, then (3 0 ex is onto. I (b) If {3 0 ex is onto, then {3 is onto. (c) If ex and,6 are one-to-one, then j3 0 ex is ohe-to-one., (d) If j3 0 ex is one-to-one, then ex is one-to-one. Theorem 2.2 A mapping is invertible if and only if it is both one-to-one and onto. 2. (5 Pts) Particularly in light of Theorem 2.2, the definition of an invertible mapping cannot be simply that it is one-to-one and onto (although as the theorem proports this is logically +5 equivlent). Thus, provide the formal definitions of inverse and invertible assumed in our,\ I. ~ ~. <- c}. " C:-)' ) I-,, ;>;) f ~$ 11-e.. 'J,r. J ea~ 0" Ii-. textbook., ~ 1 ;.. p od--- -;::: 'J.--- d, cl cr -- I... ~-r Ls V A tvl~l?pi "') i ) II/',!l,{.L1-1r) 1e -.1-1L.c 1V\'Je- ')e. e ~.\~\S
!. J ' ft, If '!'\-; ~I ""J i-. <..1.' l.".r.' C-'v-d ';). 0 "" 1 rr~(,,, l ' J 1':"-1.,) ~,- f~ t-..,..~ (),... 1 4o.... fc-' I 7 '" S,~.u. ''i) - -;,. e., - ' I r.'- '. >\., 'e-\'\'r.p...e- h Q, \~. ;...J -,. I J.. If" "" 0 ' r '~)) ~ ~l!'h' 0.. 1 / ft.'. 1(..I.!(e.".',Pv/x" S '~1 '< i' t. Qi) 1'rz:,S''''N,~~Q''.,..U.. t :;'o! i (>--<-- h ki_'l.~ ' l,," " ', J.,;.. l. \J:)" o,c..l.., \"'r II J' ' U;, ce--- "ef " ' ' L \ 1\ O S...,- ty'4 "1 ""-v-.'b- to }V'" ') : \ n I ', ~. ~ I. '\ 3. 1(81 ts) Assume that a: S --t T nd /3: T --t U. Use Theorems 2.1 & 2.2 to prove the following: (a) If a and /3 are invertible, then /3 0 a is invertible. (b) If /3 0 a is invertible, then /3 is onto and a is one-to-one. A.:s.s,~J".e '"")..". t<?j /: ( -!. I ' \ J'!Jl ~. ~./,-" +0 ~~/'.,. -')/\{ 'r-o <QV\-e ~ +0 ~ Ov: -c :J..v.,..~ c A'.'~ ~..' I<~' (.. A -,,,r..,,i b~ ;.. " ~v\':'ij,i I..N~, (l \, ~ [. r '~' tl \ 1-4. (8 Pts) For each equation below: Does * define an operation on the set of integers? Explain your answers. 4b.
Name_...:... l'_, _ ).:...:._ ( ''. _ 'V o.....;.-'-"e \ ~ =---.J('--_ Page 3 S. (8 Pts) Let Q = { m 1m, n are integers with n =I- O} be the set of rational numbers n and * be the operation on Q defined by Mb = ab + 3 for all a, b E Q. Sa. Determine whether the operation * is commutative. If not, give a count~rexample. i a _ ::'- ' JU~ ",.}. o S---..J b-::;. ~ \"'e"~ s-to. ~,> CO I\'\-"'\.Q. ~ c~,; y\ L.'0f' IS" I6u C;v.~.I')i.\l.. 0... 0.;(. ~ c::o. ~~:t.. ~ ~-.\- ~ J)' -, 1'\ :; f' So ::>,)~ d b :x-~;:..i >f- ~ ~ ~..,.<;. '5 ''1 S t" ( ' Sb. ~etermine whether th,e operation * is associative. If not, give a counterexamj?le. >f s 'd.sc,00~~ /:- 1\ o-. ~ {~ ~c.') -- ~ -l"b/,' C,.i..)(.ad" ~ \ b, c. t:: ~. ) +/j _ \.~ ~ 1:;~?,,_..!o 0. t 0-.. &>*c..)~ ",'.. a..r- ~-ts ~o.. l bg -t--o -j-? ';::: a...b'-+~o.. rs' ( q \:;,) *c. :::- (0lo-r~ G- ~(~'o n)c.. + s ~.3bc. + '3, <=- '3 Q", h'e....>- 3<: ~ --; -4- ~!a \'-' S~~ ~ / >A.) no1- ";.). '3 ':- c. ~~ '<I.e.. 6. (9 Pts) Given here is a table that defines an operation on the set S = {a, b, c}. * a b c a a a a b c a b c a c c 6a. Explain why * given in the above table is an operation on S. e (. '4,v.<;~_ h-. ~(e ~<; [.\01",,' <(.. / {f 6b. Is the operation * commutative? Explain. -f= ~) V\ D 1- C:..o...,..,. "'- ~ Jr., Ie: O. y b -:f. b- 0. :e. CA* b -;:- 0-.. 7~ ~ y. ~ ~ C! 6c. Does the operation * have an identity element? If yes, which element is the identity element? ~l~,-th U~~ ~. I..- ~,Je,.V\.f,1 d v~e.~-'r ' r, -.fo ':~. ','~ JS' ;/, ' c ~'\, t~... ";). J~.
(c_ ""'.'..(f1' L 8. (8 Pts) Explain why each of the following sets with the given operation is not a group. 8a. The set of all integers Z under subtraction. -:z, L I...l. oeul1j:.e j I j '3 Q---- \.II. V\..~ ~ ( SV\':.H('C C. <"I.;) '\ 1'-: V'II), \ - '" -\ " \ ) 8b. The set of all rational numbers, Q = {m 1m, nez, n i= O}, under multiplication. (n J n VV\ ~ ttf::\ \A)1".Ue.. n;;/:-o \'1 \ \ ', ~ 0 t- -; L/1QV'. ~ 'C...--' IJ.?I. I Y. "f1 &< ~ ~ :~c:: o;, ;~.: ' :ro.'~l~u: "A \h, \~"~r ~ ~. VV\ ~OV0e"Qj v. y V, 9. (4 Pts)~~ or False: If G is a group under operation *, then G has a unique identity. / ~ r False: If G is a group under operation *, then every element of G has a UnIque Inverse. -1-1
PageS 10. (8 Pts) Suppose the partially completed table below is a Cayley group table. Fill in the blank entries. * e a b c e e a b c a 0\ b C- e, / b ~ L- " t{ c v e- O( b 11. (12 Pts) In parts a and b, write each permutation as a single cycle or as a product of disjoint cycles. 12. (8 Pts) The positive rationals Q+ with the binary operation * defined by a*b = ~g form a group.. + a. Find the identity element of this group. \j1 de.~-~, h~... (J~ ; Je~~~ -I\.. C( {.P 1..$ e. ~ (Q S~~ ~ ". r.;, I) - 0 \I.0-0 p, fj\~ I. \ 10,., ~ -.. " '- -. ~ c~<o. C :'C>' f' ~! ~ 0"" \0"10. ~ ~ (v 'J..''-'f '1 t;- fr '" r-~~ -= l\, y,", <.1e 104 7 4 9 r Pf ffl 10 w( ~S-\/~ e --- 10. -r '& d h. f h I 4 <?.n (} _l.~ "; 9~ "\ \' ( e'l.l 0, e:. f;jf- \ 1L~(i- ls ~, b F In t e Inverse 0 tee ement g. '!:::> 0' _ r y, '\ "1.--'\ v~... b ec;t ~ o ~A~~ b ~- " :: (J..- 10 -:::;- e I
Evaluate 0:10:2 = c<.. / ) Evaluate a2a2 = DZ I Evaluate a.la 2 = 1')("1 True or@it is the case that al and a2 are inverses. 13c. Complete the following table for H. al a2 a3 arj al 0:1 0:2 0:3 0:4 a2 a2 rj I cz LY Jvl.. a3 a3 0(1{ C! ~l- / 0:4 a4 cf ') d- )- Cl I List the elements of the following subset of H.(Hint recall that permutations are mappings) 13d. {a E HI a(l) = 1 and a(3) = 3} c{j t.,/ -,.J, '-~, - -\