Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
TEST #1 - Sprinq '09
I
I
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
I
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
1
I
(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)
I
I
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.
I1
(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.
jJ ;f
-- - - -
-- - - - --+<--
-r
.. -
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>/
A has a. S';vnal/esi ;\"/fe /RJ.?1enf.-