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MHF 4102 - AX. SET THEORY FLORIDA INT'L UNIV.
TEST #2 - Spring 2009
Answer all 6 questions. Provide all rworking. An unjustified answer will receivBEGIN EACH QUESTION ON A SEPARATE PAGE.
TIME: 75 min.
asoning and show alllittle or no credit.
(20) 1.
(20) 2. Simplify the following ordinal ari4hmetic expressions asfar as possible.(a) 4+w+w2 (b) (w+2).(w+3) I (c) (w.2+1)2.
(15) 4 (a)
(b)
(15) 5. (a)
(b)
(15) 6. (a)
(b)
Gi ve the
operations:Prove that for
(i) K+11 = l1+K
definitions of thel following cardinalK+11, K.11, KJl.any two cardinals K ~nd 11,we always haveand (ii) K.11 = 11IK.
Define what are continuous and stri
J
tlY increasing class-functions from Q to Q. [Here Q = c ass of all ordinals]Let A be any set and P(A)= the power set of A. Prove thatA -< P (A) .
Write down what the Axiom of Choic
l
says.Prove in ZF that P (w), the powe set of w, can be
linearly ordered. (Hint: The set (;Ji well-ordered by E.)
(a) Define what is ordinal exponenti!tion and what is the
cofinality of a limit ordinal A.(b)
Prove that for any ordinals a, 'YJwe have (aO) y = aOy .[You may use any results you need about ordinal
arithmetic except this one, of co rse.]
(15) 3. (a) Define what A B means and write tlown what the Cantor-
Bernstein Theorem says.(b) Let Q = Set of all rational number; and N = Set of all
natural numbers. Prove that Q N.
- --
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So lu. f I()n s; fo Te.st -# 2 /
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and a( A '" sUI {C<'~ "I'd 1 i! Ii IS' a /;MII or'/na/ /
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So (/ Ik -resa/I- is Jrue /err -0/ ;f 41.1/ be !rvle ;;/1' if-I,
!;na//J assume fhat flee -resu.11 ~ e,J;if all 1< ;1 tJbre./) is; tA-- /; m'll 6ydIPlbi I. T h.v/1A
Lc<~)-IJ = SL1{&~}¥: P<,A} b~. clel ol.~!.
:= SLj/ ~ LX~.: : 0'<A} b- 1Iw. Inl hy/~ sr f ~ ~ : J' <' f'~) =- ocf")} by de.l i ejP'
:5U1te. (b ~ IS?<. l~'J1/f ord;naj, ,5;( //' It. .~es£{/!~'~!.
hv c7tII y < A ) ~ f 'tv I II ie frue 6' /) .
By 14 Pr;"r/c.!;It:- iJ / TmnJ-+~r7lje. .L./1.-~u:h;:!}1J !k ~sull
rs hue- & ~~// tfOdmtJs. r ~;Nte IX a",i f"'MY''''
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:2 (ct)..,
/L \ LT r LA)+- C,V =:. (LJ ft,J) + tAJL t~lf- tJ 2- ::: CJ. i + <0, w
- ?J - ( If- tJ ) = tU I tJ == tJ 2
(C/J+ 2 ) ~ (tV,L 3 ) =- (fA) + 2), tiJ .+ G '.f- 2) - 3
5 ~r f@u f ZJ n ; 11<,tJ 3 + (0' 2-)+ (tJ.f 2) +-0ui--2)
- S"'Y ~ it) 0 11 + 2 : n c t.J 5 f- {J' (2 f-li)) -+ (ZI-LJ) +- 2
W. w 'f- ?0 f- tJ 'f- t.-J +- '2:::: vJ Z -I- t() 3. --{- Z
r})
:=::::
~ -
(c) CW.2 + I) ::= CW. z + ;) (aJ. 2. -1-1) = (uJ~ + I), u;. Z + {,tI~2 + I
5t..1' {(t;J 2 +1). 1) ~!/I <:"t..;J.1 Z + ?t). 2+ /
- .57' f(tJ, 211) + I; n <' t;) ~ .17 + lAJ, 2-- +- /- (tJ tv). 2 ~ i,,'. 2. + I i- tAJ~2, + tLi"2- -f- I
3 (r:c)(j) A ~ l3--wzettU2J' 1k'fe ~siJCfn ',;lee: t;;fn hzn?1 .4 h 13
A ~!b rnecu2J' 1~e ~/j'lrc:A bl' eel/on /nrwz. A h 6A -< 13 'J/YlL-vtns A ~ f?:> ~~nc/ /I ~ 6 .
e) 'n e C:n prior~ 8er}) S' IeI n1luz.dY<; py!.' LA ~ 13 j\ t3 ~ A)~ A~ 13 ,
(b) (Ie-ci rl7 IN c rD. - SO IN ~ rt2
t:A- S I (N.J,r IV:) e c/-;cf)1) . NoW c//11yM VI/~lLdy 4'e;re J'eJ'tied /n ~ j;y mk = rJ orl '111£ IN n -c' ;?V- SD}l J )' )
k ~ 0 ,-?1c.asem = 0 lJ e-hhe
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f (f j J :::c<f3 /0). I hi4L i '.5 d-t b ~j e [t /0 VJ. S 0
K -f fA =1 J< X )0] t) it A ~i]} -=-1) X ~j V KX (1.~] =- ;~\t i:
!J(h;,e fll(X;U --?> )A X K 7 :I(otf) "'q;_eX) . ik j is '"
iy/ec.f?o/). So k, /,-,i == Ik X /~l / :::: (It{)(;' J ~ jtA. I< .
s- {(:0
(f)
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iel f/: A,~ Pfl) be ?Ieh/zed by r; (.)=- la] - TIUh't. rr IS
clearly- c:?iV/ IP~jec h'on . So A ~ P{A) , No..d S"7}pcse J)~ ?fi).
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