TEST #1 - Sprinq '09

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I FLORIDA INT'L UNIV.

I TIME: 75 min.

MHF 4102 - AXIOMATIC SET THEORY

Answer all 6 questions. Provide all re$.soning and show allworking. An unjustified answer will receiv~ little or no credit.Begin each question on a separate page.

Let (A,: in) be an indexed family of IUbsets of the set X.Prove that ,

II (X- Ai) = X - ( U Ai ).1

iEI iEI I

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Let f: X ~ Y be a function and suppose t

t

a t A, B ~X and C, D ~Y.

Define what are f [A] and f-1[C].Is it always true that f -1 [C - D] ~ f-1 [C] - f-1[D] ?

Is it always true that f[A ~ B] ~ f[AI- f[B] ?I

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(20) 3 (a) Write down the Separation and RePlacemenl axioms in both theirordinary form and in class-form.

(b) Write down the Union and Power Set axiobs in their ordinaryform. Then translate them completely in~o the language of settheory.

(15) 1.

(15) 2 (a)

(b)(c)

(20) 4 (a)

(b)

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II

Define ordinal addition, multiplication,

{

nd exponentiation byusing transfinite recursion.Prove that (ex+(3)+ y =ex + (f3+y) for all or inals ex,(3,y by usingtransfinite induction.

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(15) 5(a) Prove that if ex>O and (3<y, then ex.(3< ex

~y for all ordinals

ex,(3,yby transfinite induction on y. [ou may use the factthat "." is left distributive over "+" a that "+,,& "." areassociative, if needed.]

(b) Hence show that if ex>O and ex.(3= ex.y, thE;tn(3= y.

(15) 6(a) Let (A,<) be a partially ordered set. D

~

fine when (A,<) lSwell-founded and define when (A,<) is we l-ordered.

(b) Write down what the Foundation Axiom say. Use it to provethat there are no sets y and Z such that yEz and zEy.

. NHF - A:;((omalc s:e t T/zeory/

,SollAflons to JestiF/

'Or/del /~/ernd /( (jJ'1 it/.

Spy/nq, v

;;2009 .

+-

/. j~i a€-_~{J (X-Ai). Then Cl€- X-At" Ie, eaCA t'c z.~ So Ct.'~ X ~leI /'

an d ,?{ 4- Ai ~r each ('e I. ,'. atE al'ltl a tJ f~ A/.

I"~ oc X-(U Jfi" tie P1 ce /7 (X-Ai ) c X- (.UAi ) -..(7f)H- ll€--I ) 'Z~T'- l€I

Now let a e X -OiI AL)' T~ a 6 X anti 0_1; (i!l:z= Ai) ,

50 Vie X and ati Az' /d;r Cto/_'-e I. -- . '. .a ~ (?<-Ai) hy ~dt:' t?J. -

a G-t£1(X-Ai), HenLe X-(iYr Al ~ ifr (X-Ai) -". (¥-¥)

;::'yom (1f) ~ (}H-) ;f;; //()WS -Ihtfff . f7 ( Ai):=: X - 1.1) Ai,)} le I l;[~1

...

2 (a) _/[A] :=Ilia): t16A} ) f-IC} = faX: 1(t7)~ C}

(6) +!- YES. Let a(; i'I[C-])] - Then (d) G C-l). So /(oJG.c

and I{(t)rf-fj. ,'. aG_I-/[c] c;,nd a I-[})] , nus

- - a e IIc] - /-rJJ]. ~ce /-I[C-IJ] c ;-I[c] - t=[lJ]..

~) _NO. Le-L X= r-2/0/Z)/ y= [Oi'/if} 7

jA_=10l2J, B = {-z}

?tf/ld +.' )(-7 Y he de hJ1e~( b.,7 +(x)::; ,2. ?hen

PIA -8] -:= - 1[[0,21) # fC;f¥] 1; ro} = {OI~- r~J := f[I(),zl] - 1[r-2J]:;f~]-I(j?l.

3 (a) -J€jJaTal!!.~A_xIOr1!: It t;;(x) /s Ctf}Y ;;: mU/&l 0/ itJST w/It one,

!tee varl/!11e X &tnd f} ~s a s:ei then [x r;;A-: ~()'o3 is Cl sel.-

Cltl5f'--hr-_~; /1 h is a class i.; II Ihl?n ~ (J A IS a s~f, - -

,/?~placerneJ11 Ak'(crn: II ~{x,y) Is q !it"c ol1-~/e hrmu/a tJ_/ !..IJST /Z- - -

i1 IS t{ g~f 7 then {b; (p(a,b) holtl~ /'or af sf one ac/{} ,~ a ~~d.

Cltlss-/orm : I/' .3 if tJt cIaSJ'-IZa7cho11 ~ A- CJl.sd IheJ1 :J/!I] /s a s~it+! . . - / - -'

(h) . Union AXion?: l-t A is a set /' $zen A is' a s-et.

CVXj)(9X2J('rIX3) ((X3 6 X 2) ~ (3 Xtf) (036 Xif) 1\ (XL; e XI)) ) -

Pot-1Ier Set A X/OWl: If A /s a S'et then IJ(A) is a. set.-J

-+-ii/Yt) (}X7-)C't/X3) ( ('7(36';(2) <~ (\iX1) ( ex 6X3) ~0'16XJ)))

Lf (~) Ordinal ?l dtlhOI1 1 TYJ t-tlltflicalt;nj ~ w/onet?l/a!/on .c:::tre

de.hne d bl ;;aran1e Irlc recursion on J / as -Ib //tJN5 ~

(i) .l~) iX~12 ~ - ~ r (}) eX+ Cf+}). == (0<-t-p2 ,) l~)- 0(+ A ~ SlA.t [0{ -l-fi ! f ~~J-

Ll/)jH(qL'L q(. 0 =- -OT Lb)O(.~+-L)=J~#)-f 5 -;- -(c) . A := Sl/IP(ex'f: ~~).1 j al1j"J. 0 . ~tl P) A $

((II .tqL_ri =- i,- ll> 1 0( - -=- (ex ,iX 7 (c') ::::. ~/Al [0( : ~ ~AJ '

- - ){ere- _c<-LS fhg !£{rqffl~fer _a!1d .1 is //m; t o-rd/ndl.t+ - - - -

- -- - -

{b):~ We w;/Ijrove i/ze resa/I b;Y-jan:U1Jefrf frO/JS't'n!4 f/'nduct?cm o~ y. -

-HiSo_le.j 0( and.~ b!!- ar6/lrtClf7 b~f It e/, FoT Y=o/ weh411e

t+-- -LCX:-f-f)+- r := {o(+ f)-f 0 = (0(+) _by UJLtl)

- H- - - -- - -- =- (0<-+ (fir-O)) = c<+(j -f-r) Py (/){q) arfd/I!._-S;0 -fh e. res u /f is Irvce Idr ;r:= 0 .

1 -- -- ---

As;sume ihalltie resull is .J-rue /drr r- -- - - - -

- -- ~$o LeX+-f'J+ct.f-i )- = [(0(-+ j{Jf- Y} + /

- - -H- - = (D(+~ r'11) -I- I

:= ex: -I- ({j + r)-I)-- - - -tH-- - -- -- - -- -- - ex -+- Cf3 + (r+( ) )

SOl! Ide resulf is; I-rue hY r if -vJ I,f- - - - - - - - ) - - .

-- -j 't-hn ?!.lIy a s~ume fharthe Tt?su/J If!; -Ir.

~ t-A--;~ ~- I.!mlf ordinal 7/ze}:1 (0(-1f)+ r =

- ~ t-S_o (o(f_f3) f i\ =- ~'1' ffO(f{i2+ '(.' r<-A

= S CAt I0( fif f '0): r< )t,

- - c:< + ~}:l-f1f +Y: 1 < ~ 1=- 0( f- (f + ~ )~ >- - - - ---

- -<f+-5o- -;/ -!he res~/I /s Irue (if! ~~ 1l ~ - /-:.

-+ J~y ftL fj-1/?_Ct)1eC)I 7ra;?S'Ii~;fe In/ucjlon /

all- r. StV1ce ex:an,! f ttlt:re Clrbi!n:r//

- -+ ~// OYJ~~q/f~ f, r. -

- -. -

hen (C>(~/!,)+Q-; CX+lfT!) --

b)1 (/) (J) -

by Ind. hp?

j,X (i) (1.) ~! Cti n

/;y (j) (J) "nee ~tla/--".

le frue k '1f-I,

e k a/I r <.A uj;;eye- /

D(t(p+ r) 1m-- a II -Z<-) .

/;)' (;) (j;)

hy Ind. h/?

by (1) (c) C<.tfa/11 ~

by (;)(c) cn?O? a/din.

1/1/1/1 /;e fru e ~/ .:1 .

tte resul!s ~/!ou/l ~the ?"eS lA/ls hllouJ-hY

(~)(';rl1) :(fr)7h/~ workr; because r.sU; If+u: r<A)' J/s a !;mi! ard'//7C1(

0<.+S = Cf~ [0(+ C: t<sj = 5L1f{rx.-+(ft>Y-°):p'd'<sl =- sUf{rx+{f+o): d"<()}

'S'(a) We wIll trove iiLe -resuH b.1 parameJI'/;'

So iet ex ?(j//4 f b£?- diy6. bul hxed

value ~I Y wIll Ie f+-I. So we slayl

h>Y Y::::: ff-/;I we hal/~

0(. f < ex,~ + ~

= 0(. Cf +-f)= o(.¥

l-bu-Le ~ reJuif /i ,frue j; r f +-/ .

A?:,su"rte-that Ik rest~/I I~ !n.ie '

T~ f< 'f ==;. eX. f < eX. '(

"'< MeA 2--iIher f3 <' 0' (!)r f = r.

ex:.f <" 0(. t < eX.r + 0( =h- Anti 1/ f=: Y) ~ ex.! =

- H _So f< of-I ~ of" < rX.(t+/).

frue kr- d; Ii IN 1-/1 be !"'/A£ -tr-

++_hna/(y aSgUJlne thai lite resa/I ~

f4 A if; ~ /;h7;f"n:llna/ ~ f+1. S';-j°- ~ !~ f3 < ~ ;;, SOn?e ~ ~ ~ .

0( " < DC to because-the

::::: s~ { o(~ y ~ p< ~J~c£ If-/k reS'dl If' frt-t.e- -kr all

becCtus;e

?rang/'. Jncl on r.

S';l1ce p< 't 7 ik fear!

Ind. a-/ y=,f+ / .

(j;) cJlihe del tJ I II 1/

Y,- where ! ~ f-rJ ,

sU;;°s~ t < ;I-f / ,

f<Y} f~ex. UO+-J ') j e c Viuse. ex > 0. -,' ,

cXL0' < 0<:.'1+ rx z=: IX .(Of~) .

c.e II ~ ~eJu/1 /s

-fl,

/ne k t:Z// ;Y.( ~ uk)

qZow Ituaf- ! < ~ ,

So

7t'!?Ju/l IS f-YttCe k 10

eX.::) .

1<~ If IV;/! Ie f?'u£ ~ I).- /

BJ IXe Ibrameloc Trans4n/4 7/2dlt~h~! ?r;~c://t/~.7 t1a resu /1

h/lcJ~s £r all cr; t~~d ,..

(b) -,4 s5uvne lAat rx >0 vf)'!d 0<., = ex:., N"tu 5j/ose ;!hat

-(3 =f d, Then - ~Ilkr f <: Y' 07 < t- 13 vd If 1< Oy

- f~ ex'f < 0(. r (by j?arf(a)) - CO nJrad/c///lf0(,/ -=- 0(. a'+ - IJ /1vi if Y« f,; theh 0(. r< D(. f j;1 ,? crr f (a) Vi!- a j h )-'+ _?o Yl Ira. dlt~.i, n-d eX'-f -;:= of. r . S t' n e;ji-t!l case w..e-do!

a ~OY1frad,'c./;'c;YI. J..but:£ sf have 1= y.StJ

Co(> 0) 1\ (<Y.f = 0(. r) ~ f=

- {!oJ...Fe" n de.!,;'", /I Xl""r>: / t: jl ,s a"l ]""'71'e-",)')- sdJ I~-!ht:2ye is an X e-A s uclt thaf X Ii AI =- f2f .

S'lAlf-tJse tlzere edist se/s)" ane;( such ftai )lG-Z

Ctj'1-d Z~p/' LG!( 11.:::rx ~J -~ IAe z- ~- .1./1 A bee-,/!use-

Ze; Y an i ZG A. IJ.kD Y6 z nA e-c.aVlse YG Z C01d Y~A.-~+-- - - - - - - -

51f1~e A has c?//11.7 fWtJ _.ei.!:.I/Y}enf~J n n?B)t Y- <:\;2!:/ I}£ l/tJ~~.

l/iaf fkgre /s ~o - X c: If 5tlC~ e:ii X (J A = >Lf. 15 ~'f.f

-thIs LOVZfraJicis It.£ ~ndaj-,on /I 10m. /lerzce ~eye are_. - + - - -

?1~ s-e-ls y and Z s t,u:A tJzaf Yelz aJ/J d Z~ Y .

(E xl-ra) (r..) ) ; S ?t. ?'YI;11/m~ / e/e/41en! tJI l3 A ,/ It-ere / s 11~

H~- X /n 13 w/IA X<: 6. (x ctJuld~ '11o1?-com?art?l~/ew?1i61

J¥-,,) b is 0<- 3m a lies! e/-eI/l1.enf 0/ f3 =::-A- 1 f 1'07 t?71)1 XG f!:,I

X ~ /"' o.

-+t1A- st:!6sei B Ca//2 hqye --many nun ~4"/ eJeWlenij

c:.an have 0..,217 ~Yle s1/Y7al/esf e/evn elni.

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b(tlj (A I <:> (s well - founded it ever} fJ1tJl1-m7fyS?t h.r-el 6-/

*. t- - -- H Jl- has &<- .-rYJ//)I'n?/ eleen .

H<A 1- <:> I- well - ern/ere d/ €ye71 ??CJn_-£/

su6sef e>/

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