- -, .- - - ,-. -

tMHF 4/02 - Ax 5'd 7hetJy~

L '7£5 T -# 1 - SprfJ"!1 ;2~tJ'r,. v

,~Al1swt!Y 4/1 6 7tlt?sl/')J/jS, JiAsl/(y all,

Gill

?/{)y/da. In! / lIYJ!v.TI/V/E: 75" mlYl.

- . - , .-n'

"I JPlA r prfn .Ji/1/eYS ,

,(J5) 1.1tet (8 i )l€I f,e Cl~m/Iy t>/ si/lbs§ls CJj/ £. Prove

l~hat If< - (O- BJ) = .U.- fjR - Bi )

. ~ ti~ I I. tG 1 {!,,i.'r"-

. (;kt;j :2 ~4> ~eh/if . ()rJ ;J1t// lYIulf¥/caf/oJ1 f:,y VISinJ f ml1S IJ ;) jf ere (jAr,$b 11

#)frove fha! tn' any oyJ;;1t'df I)(l'}' ~ '(1+-1') = ot} /-1)(.',1~ (You rYlay ~1setAt! Act !J,?i! tJt"d/I1/l/4t1dlfoJ1 is aSfO[/p:t!/t!l?)~....

{~(}) 5l~j tJr/fe dowYJ fhe~eft4ral/oi1 Axiom ~ Me R~Jc:lt:el11€I'Lt

;"'" Ax/om in /;ofh the ir tJrd/r1ar I ( C/ptSs ft r ms I

~) tJr/le t!oi/l)n fhe Pol/Jet' tet Ax/p/y! t. /lif- FOundaliorJ

; Axiom In Melr tJrd/;1ar.1 -form, TAen J-ras/ale !/ierYI

/1110 lAe IVl7fl/(,ay_e 01 sef JhetJ';Y/ (LtJsr).

uS") If.~ie t V~ co/le,;r/oY1 0/ a/I sels & S' ~ rx: X has one eJel'riC'nfj

(a.)Hljs/~j fhe kef ffiat R =: {x ~ X( xJ IS' ?'Jot fA 5'e~?rovelWic4.1V ~ S are 71ai se-fs,

(/;»1~PrtJve1hi/!t g is a £-/ass.-'"

(iD 6-' tq) JJeh'I1E lJ h~;zl ;:; Cft tV ell - arde red fe f (A) <> .,- --- (P) /1 (A;<'> t~ A _vye/l~ortlered Jef and f: A-)A /r an

L, InCreaSi/lj It-'Ylcf/on -' fJ?Jye -lAat x~j!{x) tv-ra//xtA.

(/s-) ~.(4) lJel"ne f/l/f,iPlt i~ a frVlI1J"I'4've set ~-t;fJhat l~ an ~rd/na/.

(p) S(Aflose R is rel/f/iiVe anti aRb J hRc ~ t!.Ra .:.. Prove that R /$ an ett1; va/e,lce 7e/~{I/{)/1. I);? Ii .

C/5

:,M}-IF ifl!) 2. - It>:. s:'et Theory,

_~SOI0f/(}J1J; ID Te£l # /

FitJvida Intl/ 1I11/v.

$'pr/~nq 20(J'f. J

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,~.S () X G /~ Ctnd ('Ji E? I) (x.-1 B1)

/. (:;}i c I) ()( ~ iR and X tJ Bi ) ,

l', ( 3iG L) ('x€- /R-l3i ) .', X ~ U (/R.-Bl"

.,., - . le I ).., 5 0 R- .n Bi ~ ~U (lR.-13i)

. t€I 'lG:J

NcJVJ /~d X6 IrTll/(-13i), 7~ @1E-I )(X6 IR-I3;)",S-o X 6 !I( Cf;1~ X <I8 i firr StJ#11e i 6 T

,-- ~ ' . X G JR.. C}/Yld (3i € 1) (X fj B,i ).-,.'- )(G Ie and 7 (Vt'Gl) (z<e- B£ ),-, . " )( ~/I: a nd X1:.fl 8; .' S;D XG /R-.f) Bi

. tIEL u-I

,. ~v~ ,.iI - fIR. - Bi) c: ~ - ,,(J I3ltb-LC !l!!:I

IR - ((J Bf) ~ ,V I (/R- Bl)(IG-I 'lE.. nus

2. ;(4) fin- ea~1t cffdo,1t;1/ f w£ /€~ f:r b; fmns/';v IeL 7eCur, J (JYI 0Y7 ! Cls h /It; i/JS : -

;t U) f< () == O} 0;) f' (r+J)-= Cf- r) f- f J Vind

" ijil) f-L s~ {f' '(: rd} II ~ 0 a //t11i1 tJr~i1tf1l

~) WR WlIII'7Jve /Act.f 0(- ~+r) ~ ot-p f- 0(, r b;-

" frt>lnsf, nde //1,1«cf,'(};l ~n r, Si/N e d0= tJ, 7hi' 11

£x. (f 'f- 0) = ct.'. P == Dt. f3 + 0

~ ()(. ft + 01.. ()

5' 0 Mt /j'esVtlf IS fTMt!- fry' r = ()

"

2.?71r°S:.e #ie 7'e..r~df is f7Me j;y

..- /f kY O'fl. /Vol-J

oC {f +(r+ I)) == ex.(iff-r) -f-/) /J.ec. aM liar; is tf.fJ ty.

.::: ()( . ~ t-lf) + 0(

:::: (oc. p +- IX. 0') + ex

== D(.~ -I- &<. r-1- o()

= {)(. 'f -I- d. .((/+J)

Clb

r. tJe VJ /11 jT01/e

b)l (iJ) a bt7//i'-reJu/f Irk£! W r

Vtd'/, h'on IS' t:tS'£{)(.

b1 (ji) aho vt::

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()/. if> 1- A) ;: 5JAf [ 0(. (f +r): 4 < /\ } /Je[ f +-A ~ IImI t vrd.

-::::: svr rex'f + rX.r: r< A} reJu/f Irl1e frr I"<A

- f¥. f .j- S&If rIX . 0: 0<A}- ex.f.f ()(. /\

,So b7 Ik '7i",?{I1's,f1~n/fe Induch~fY7 Irll7C'yJk .I lie

-, 1esu.lf ~ frVl: h a/I r, Since D(~' f tVer~

__arb/fattY j If 0 !-YVle fry ~/I 0<.1 f ~Yld d,

~(tl).S~f' !lx/Of11: If I/(x) is any IJ~ mula. t/I L-.IJ.s.7. and__A iJ"' C/is--e-i 7!ten {xt=A: f/{x) IS' frlAcf 7.r a .{:£l

--Cia sS' FtJ011: 1/ £:; ; sac la ss <R. jJ is a .r-e f #,en .A /J /II S a set.

- R~p/ IIx;~m: II r;(x, y) /s L:?11(!J;n[f/tJJ1-flle hrma/a0 I L. (). s, Tan d 1-1 IS ~ re t i j,i?11.-- J

- r b; ft (a,/:;) /~ frVle- fr~. e<f lettlsl n1c? Cl€A] is' a s~l.

(~Ia 5"S'h rrY!.' Ir c-.1- / san / c/ a ~s.1u )1 ('. f /0}1 and II.is A ~-e f I iJ,en J [A] = {cJ(a) : 11€ A} ls ~ S'ef .

(b) ?ower S!{:l !l-xJvm. If 11 /s Vi S'ef 1hen r?(A) /s- a sef

pc C/?

3 (b)I~(Vx,}( .3x 2) (I7'X3) (X3 e Xz 4-4> (Vx'fJ (X'f")(3 -.. Xif ~XI ).'IV- . -

i~F"u J1dVI liD 11 II Xj om: /1 /I- :"san y 710 n - G'1YIj'1y S'.e( 7iu("li0t is.is X'€ II su £:6 tA?lt Xn/i := ~ <

1((Vx,)((3xz)(X2-€x/}-? (3X3)( X36 XI &; (VX4)

jl .., ((X'l (; XI) !( (Xlf 6 )(3)) )))

If.(~ Suy>pse V /";>C/.se-.t, Then. iJ,y' /J;,e s;-r:"ra f/o ni~ j))( i'o n1 ) [J( GV; X rf x} 1-tIJI/ be a ~~&t, 13 at:! fx' G V: X ~ x} =: [X: X 1 X} = R /S;1 tJ t a s'-e- f - -

50 we !lave a t:-1J1y1f~c/;ch~i1. !/eMce !lis Y/uf- a s:e/,

~~ /IIolAJs~f()se S ; S' a s'e-t. The 'l lIze un/tJrt{)Lxiomj V S .w III be a S'f!;t. 13ui u S

~ u [{X3 ~ X lJ' 4 f:bt} =- f'X: X j:; a ~~i} =- Vf/Jh lc'h IS' 'Y1ol a $e-l - sa we hCflle a no j;{ e y

.1. tIJr1 T,-ad erl/ol1./l-bnzt?- S ;s '?<1tjf a ~'et-;

(~) We j' //u.t ha ve 10 Ilnd a -/br n1t4/a 13/ L. IJ.f. 7

{A!h/ch KellS tJ.e- ~4me fillY ~S'''x J,,2J Me.e/ei/I'lt?ni. kl r; (X) ,/Je !AL1df'mula

(3 x, € x) /\ (Vx'2-)(rtX3 ") (x2 E- X &. X3 r=- X ~ )(2 =-.X3.) .TlzeV1 S =- f X :fl(x) ;s frue} and so IS' a c/a_~.[,

5:tit)/I we/l-crYde,ed sd (IJ)<> /£ a fY'rl/?i/1y cfYdi!red~ef

~L <A}<> /n I/J h leA qn 1 rJ();1- erry7 S i/lbJ:.t:t hAi' ~t.- 9ma Ilest- e/emerif ~

rla Rect?ttl lAt-tl I: A~ /} is /l1cfe'?lJinj ;1 X <::y

:. ~. /(x) < 1M. NoVo) fltyrse :. (VxGA)(x ~ I(x»);~ kt B = fx€-I/: . ftx)< xl. lAl!11 B:f 1J. l' b

! B hag a s mallesl e!t?pneJllt I b S ~/ .

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Ii Ik Sma lies f e/fJ!Yl.~~YJi l>f !3.J S tJ (.Ale he:!v t: ap .

~f tf4nIYAd/~f(pn .~.IJt?nce. Lbfx£A) (X ~ /!(;<) ) ,1\

J~ .-- n- .- - -. ..

0{ajlA set S is frClf1~il/l/e if a~h~ bGS ~ a€S.-I'

hArz tJrcl/Y1a/ is ~q.-~_j-7.::t/ls11/ve §el..A ~'.AA,;,hthat

t~(AJ~> is Vi 51!'.!~lly "vle//-ordey~tI S'~f,

cl3

IIj1. -~_. .

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~l-t-Jh Ich / S' ,-yel@jy'f-/ fran~ff/ve. Cfl1d symPJ1elr/c.

1...N"w we alre4'l:iy .knoi/v fAa-l R /s re//t}?{)ve.L

JtS<Y1'°se a j( b .]l.t?-YI 1/2.6 buauu: R i ';;It re I/exive. ;/-~~e a R /, and hRb 5:/n~e.

11:CLRb6:b Rc~ ~-R4 ;1 .Allows Mp-I '6Rt'i. So, . . ... m..''''_. ) --

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1:' R I'~ S1m #1€Vfr t~'~J!i

Ir'0,na/~ $'rloscf'lRh qnd bReo 7hen cR~

1t~m vJha- lAIa!;:#,vel1 Bur w~)ud. s-JuJvtled 8-

F.t-'Jas symmefrt'--"-" ~ al2c. So C?t~~It. t:JJh.d /; /(,e- ~-~C{ Rc . Titus t( /' s: I-r t:9fI1S /'1/ l/'e. ,

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jl/~ f( i £ .q n.e1V1 " va./el1£e yej,a f;0J1 .i.

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