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5.5 GEQBA ACHAF[ Q EHN BL[CW EH^ WFQVI Q(e) Wfietbch rcl E tc GB` E, G erf twc (hch-flpty) sfts, tmfh ehy sugsft c` E G bs aeiifn e rfietbch rcl E tc G.

B` W bs e rfietbch rcl E tc G, tmfhnclebh c` W < {x ? x E, (x, y) W `cr sclf y G} ehn rehjf c` W < {y ? y G, (x, y) W cr sclf x E}.

B` W bs e rfietbch rcl E tc G, tmfh G bs aeiifn acnclebh c` W.B` E ehn G erf bhbtf sfts achtebhbhj l ehn h fiflfhts rfspfatbvfiy, tmfh tmf hulgfr c` rfietbchs`rcl E tc G < 7 lh.

(g) Wfietbch ch e sft E(b) B` E bs e (hch-flpty) sft, tmfh ehy sugsft c` E E bs aeiifn e rfietbch ch E.(bb) B` E bs e bhbtf sft achtebhbhj l fiflfhts, tmfh hulgfr c` rfietbchs ch E ypfs c` uhatbchs5. Chf-chf uhatbch. E uhatbch ` rcl ] tc R bs aeiifn chf-chf (cr bhofatbvf) b` nb` frfht fiflfhts

c` ] mevf nb``frfht blejfs bh R b.f. b`` x 5 ] 7 ` (x 5) (x 7) `cr eii x 5, x7 ], cr fqubveifhtiy,`(x5) < (x 7) x5 < x 7 cr eii x 5, x 7 ].

7. Lehy-chf uhatbch. E uhatbch ` rcl ] tc R bs aeiifn lehy-chf b` twc cr lcrf fiflfhts c` ] mevf self blejf bh R. Bh ctmfr wcrns, e `uhatbch ` rcl ] tc R bs aeiifn lehy-chf b`` bt bs hctchf-chf.

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1. Chtc uhatbch. E uhatbch ` rcl ] tc R bs aeiifn chtc (cr surofatbvf) b`` feam fiflfht c` R bstmf blejf c etifest chf fiflfht c` ] b.f. b`` acnclebh c` < rehjf c` b.f. b`` R < (]).

>. Bhtc uhatbch. E uhatbch ` rcl ] tc R bs aeiifn bhtc b`` tmfrf fxbsts etifest chf fiflfht bh Rwmbam bs hct tmf blejf c` ehy fiflfht c` ] b.f. b`` rehjf c` bs e prcpfr sugsft c` acnclebh c` . Bhctmfr wcrns, e uhatbch ` rcl ] tc R bs aeiifn bhtc b`` bt bs hct chtc.

6. Chf-chf acrrfspchnhfaf. E uhatbch ` rcl ] tc R bs aeiifn e chf-chf acrrfspchnfhaf (cr gbofatbvf) b ` bs gctm chf-chf ehn chtc.

Qbhaf wf wfbii c tfh gf prcvbhj tmet afrtebh uhatbchs erf chf-chf cr chtc cr gctm, sc wf cutibhfs tmf tfamhbqufstc gf usfn ?

(b) ^c prcvf tmet bs chf-chf, wf lust smcw tmet fbtmfr (x 5) < `(x 7) x 5 < x 7 `cr eii x 5,x7 ] cr `cr eii x 5 x7 ] `(x5) (x 7).

(bb) ^c prcvf tmet bs chtc, wf lust smcw tmet fbtmfr cr fvfry y R, tmfrf fxbsts etifest chffiflfht x ]

suam tmet y < (x) cr (]) < R.B` E ehn G erf hch-flpty) bhbtf sfts achtebhbhj l ehn h fiflfhts rfspfatbvfiy, tmfh(e) tmf hulgfr c` uhatbchs rcl E tc G < h l .(g) tmf hulgfr c` chf-chf uhatbchs rcl E tc G

9, 6, ^ 7 wbtm

sbnfs 6, 57, 51 ehn ^ 1 wbtm sbnfs 0, =, 5;. Zmbam trbehjifs elchj ^ 5 , ^ 7 ehn ^ 1 erf rfietfn 3

Qci. Mfrf, (^ 5, 7) W b`` 5 bs sblbier tc ^ 7.

Es fvfry trbehjif bs sblbier tc btsfi`, tmfrf`crf,(^, ^) W " ^ E W bs rf`ifxbvf.

Ejebh (^ 5, ^ 7) W ^ 5 bs sblbier tc 7 ^ 7 bs sblbier tc 5 (^ 7, ^ 5) W Y W bs syllftrba.

Hfxt, ift (^ 5, 7) W ehn eisc (^ 7, 1) W ^ 5 bs sblbier tc 7 ehn 7 bs sblbier tc 1 ^ 5 bs sblbier tc 1 (^ 5, ^ 1) W.Y W bs trehsbtbvf.

Mfhaf, W bs eh fqubveifhaf rfietbch.

Es10

=

ehn 6 3

Qci. Qbhaf [ ehn [ mevf tmf self hulgfr c` sbnfs, tmfrf`crf, ([, [) W " [ E.Y W bs rf`ifxbvf.

Ift ([ 5, [ 7) W [ 5 ehn [ 7 mevf tmf self hulgfr c` sbnfs [ 7 ehn [ 5 mevf tmf self hulgfr c` sbnfs ([ 7, [ 5) W Y W bs syllftrba.

Ift ([ 5, [ 7) W ehn ([ 7, [ 1) W. [ 5 ehn [ 7 mevf tmf self hulgfr c` sbnfs ehn eisc, [ 7, [ 1 mevf tmf self hulgfr c` sbnfs

[ 5 ehn [ 1 mevf tmf self hulgfr c` sbnfs ([ 5, [ 1) W

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W bs trehsbtbvf.

Mfhaf W bs eh fqubveifhaf rfietbch.

Mfrf bs e trbehjif, tmfrf`crf, [ E bs rfietfn tc b` [ ehn mevf tmf self hulgfr c sbnfs, b.f. ([, ) W b` [ ehn mevf fquei hulgfr c sbnfs

cr ([, ) W b` [ bs e trbehjif.Wfqubrfn sft c` eii fiflfhts wmbam erf rfietfn tc bs tmf sft c` eii trbehjifs bh E.

U.> Ift ` ? H H gf nf`bhfn gy

`(h) <

fvfhbshb` ,7h

cnnbshb` ,7

5h

`cr eii h H.

Qtetf wmftmfr tmf `uhatbch ` bs gbofatbvf, Oustb`y ycur ehswfr.

Qci. Jbvfh (h) Ift ? Z Z gf nf`bhfn es `(h) < h 5, b` h bs cnn ehn (h) < h + 5, b` h bs fvfh. Qmcw tmet

bs bhvfrtbgif. @bhn tmf bhvfrsf c` `. Mfrf, Z bs tmf sft c` eii wmcif hulgfrs.

Qci. Ift l, h gf ehy twc nbstbhat fiflfhts c` Z. Zf smeii smcw tmet (l) (h).

Aesf (b) . Zmfh gctm l ehn h erf fvfh, tmfh

(l) < l + 5, (h) < h + 5

ehn l h l + 5 h + 5 `(l) (h).

Aesf (bb) . Zmfh gctm l ehn h erf cnn, tmfh

(l) < l 5, (h) < h 5

ehn l h l 5 h 5 `(l) (h).

Aesf (bbb). Zmfh l bs cnn ehn h bs fvfh, tmfh (l) < l 5 bs fvfh ehn (h) < h + 5 bs cnn sc

tmet (l) (h) bh tmbs aesf eisc. Qblbieriy, b` l bs fvfh ehn h bs cnn,`(l) (h).

Qc, bh eii aesfs l h `(l) (h).

Mfhaf bs e chf-chf uhatbch.

@urtmfr, wf smcw tmet bs chtc.

B` h Z bs ehy fiflfht, tmfh

`(h 5) < h b` h bs cnn ( U h 5 bs fvfh)

ehn `(h + 5) < h b` h bs fvfh ( U h + 5 bs cnn)

Qc, fvfry fiflfht c` Z bs tmf -blejf c` sclf fiflfht bh Z. Mfhaf bs chtc.

^mus bs gctm chf-chf ehn chtc, b.f., bs e chf-chf acrrfspchnfhaf.

Achsfqufhtiy, bs bhvfrtbgif. Zf mevf sffh egcvf tmet `(h 5) < h b h bs cnn

ehn `(h + 5) < h b` h bs fvfh

h 5 < 5 (h) b` h bs cnn

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U.56 Qmcw tmet tmf uhatbch ? W {x W ? 5 9 x 9 5} nf`bhfn gy (x) < |x|5x

, x W bs chf chf ehn

chtc `uhatbch.

Qci. Jbvfh (x) < |x|5x

+ <

9-

+

;xb` ,x5

x

;xb` ,x5

x

.

Aiferiy, N ` < W es 5 + | x | ; " x W..[. bs 5 5 ? Ift x 5, x 7 N` < W suam

tmet x5 x7. @cr aesfs erbsf ?Aesf B . B` x5 ; ehn x 7 9 ;,

tmfh `(x 5) < 55x5

x+ ;

ehn `(x 7) < 77x5

x+ 9 ;

`(x5) (x 7)Aesf BB. B` x5 9 ; ehn x 7 ;,tmfh es bh aesf B,

`(x5) 9 ; ehn (x 7) ; `(x5) (x7).Aesf BBB. B` x5 ; ehn x 7 ;,tmfh x5 x7

5 + x 5 5 + x 7

5x5

5+ 7x5

5+ 5x5

5+-

7x55

+-

5 5x5

5+ 5 7x5

5+ 5

5x5

x+ 7

7x5

x+

`(x5) (x7).U.50 Jbvfh e hch flpty sft ], achsbnfr [(]) wmbam bs tmf sft c` eii sugsfts c` ].

Nf`bhf tmf rfietbch W bh [(]) es `ciicws ?

@cr sugsfts E, G bh [(]), EWG b` ehn chiy b`

E G. Bs W eh fqubveifhaf rfietbch ch [(]) 3 Oustb`y ycur ehswfr.

Qci. Qbhaf E E " E [ (]), tmfrf`crf, W bs rf`ifxbvf.

Eisc, cr E, G, A [ (]), E W G ehn G W A

E G ehn G A E A E W AY W bs trehsbtbvf.

Mcwfvfr, W bs hct syllftrba es E G hffn hct blpiy G E.

Qc, E W G ncfs hct blpiy GWE.

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Mfhaf W bs hct eh fqubveifhaf rfietbch.

U.5: Jbvfh e hch-flpty sft ], achsbnfr tmf gbhery cpfretbch * ? [(]) [(]) [(]) jbvfh gy

E * G < E G " E, G bh [(]), wmfrf [(]) bs tmf pcwfr sft c` ]. Qmcw tmet ] bs tmf bnfhtbty

fiflfht `cr tmbs cpfretbch ehn ] bs tmf chiy bhvfrtbgif fiflfht bh [(]) wbtm rfspfat tc tmf

cpfretbch *.

Qci. Ift F [ (]) gf eh bnfhtbty fiflfht, tmfhE * F < F * E < E `cr eii E [ (])

E F < F E < E `cr eii E [ (]) ] F < ] es ] [ (]) ] FEisc F ] es F [ (])Y F < ]

^mus, ] bs tmf bnfhtbty fiflfht.ift E [ (]) gf bhvfrtbgif, tmfh tmfrf fxbsts G [ (]) suam tmet E * G < G * E < ], tmfh bnfhtbtyfiflfht.

E G < G E < ] ] E ehn eisc ] GEisc, E, G ] es E, G [ (])Y E < ] < GY ] bs tmf chiy bhvfrtbgif fiflfht ehn

] 5 < G < ].

U.5= @bhn tmf hulgfr c` eii chtc `uhatbchs `rcl tmf sft {5, 7, 1, ...., h) tc btsfi`.

Qci. Ift E < {5, 7, 1, .........., h}.

B` ` ? E E bs eh chtc uhatbch, tmfh rehjf c` < E. ` bs chf-chf.

(U B` bs hct chf-chf tmfh rehjf c` wbii gf

e prcpfr sugsft c` E)

^mus, bs gctm chf-chf ehn chtcb.f., ` bs e chf-chf acrrfspchnfhaf. Hulgfr c` suam `uhatbchs bs tmf self es tmf hulgfr c`

errehjflfhts c` hulgfrs tedfh eii et e tblf,

b.f., h[ h < h! . Mfhaf h! uhatbchs `rcl E tc E erf chtc

U.52 Achsbnfr tmf gbhery cpfretbchs * ? W W W ehn c ? W W W nf`bhfn es e * g < |e g| ehn

e c g < e, " e, g W. Qmcw tmet * bs acllutetbvf gut hct esscabetbvf, c bs esscabetbvf gut hct

aclluetbvbf. @urtmfr, smcw tmet " e, g a W, e * (g c a) < (e * g) c (e * g). Ncfs c nbstrbgutf

cvfr *3 Oustb`y ycur ehswfr.

Qci. @cr eii e, g W, e * g < | e g |

< | (g e) | < | g e | < g * e

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e * g < g * e * bs acllutetbvf.Eisc, cr eii e, g, a W,

(e * g) * a < | e g | * a < || e g | a |

ehn e * (g * a) < e * | g a | < || e | g a || (e * g) * a e * (g * a)(U || e g | a | |e | g a|| bh jfhfrei, es eh fxelpif || 1 6 | : | < 6 gut | 1 | 6 : || < 5)Y * bs hct esscabetbvf.Ejebh ecg < e ehn gce < g

ecg gce c bs hct acllutetbvf.Mcwfvfr, cr eii e, g, a W,

(ecg) ca < eca < e

ehn ec (gca) < ecg < e

(ecg) ca < ec (gca) c bs esscabetbvf.@urtmfr, wf bhn tmet cr eii e, g, a W,

e * (gca) < e * g < | e g |

ehn (e * g) c (e * a) < | e g | c | e a |

< | e g | e * (gca) < (e * g) c (e * a)Ejebh, ec (g * a) < ec | g a | < e

ehn (ecg) * (eca) < e * e < | e e | < ;

ec (g * a) (ecg) * (eca)(U e ; bh jfhfrei)

Y c bs hct nbstrbgutbvf cvfr *.

U.7; Jbvfh e hch-flpty sft ], ift * ? [(]) [(]) [(]) gf nf`bhfn es E * G < (E G) (G E),

" E, G [(]). Qmcw tmet tmf flpty sft ` bs tmf bnfhtbty `cr tmf cpfretbch * ehn eii tmf

fiflfhts E c` [(]) erf bhvfrtbgif wbtm E 5 < E.

Qci. Ift F gf eh bnfhtbty fiflfht, tmfh

E * F < F * E < E cr eii E [ (]) (E F) (F E) < E cr eii E [ (])^edbhj E < ` , wf jft

(` F) (F ` ) < ` ` F < ` F < `

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Cgsfrvf tmet E * ` < ` * E < (E ` ) (` E)< E cr eii E [ (]).Qc, ` bs tmf bnfhtbty fiflfht.Ift E [ (]) gf bhvfrtbgif, tmfh tmfrf bs G [ (]) suam tmet E * G < G * E < ` . (E G) (G E) < ` E G < ` ehn eisc G E < ` E G ehn G E E < G^mus `cr eii E [ (]), E * E < ` . E bs bhvfrtbgif ehn E 5 < E.

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U.5 Bh tmf sft H c` eii heturei hulgfrs, ift e rfietbch W gf nf`bhfn gy

W < {(x, y) ? x H, y H, x y bs nbvbsbgif gy 6}

prcvf tmet W bs eh fqubveifhaf rfietbch.

U.7 Ift l gf e jbvfh pcsbtbvf bhtfjfr. [rcvf tmet tmf rfietbch, achjrufhaf lcnuic l ch tmf sft z c` eii

bhtfjfrs nf`bhfn gy e g (lcn l) (e g) bs nbvbsbgif gy l bs eh fqubveifhaf rfietbch.

U.1 [rcvf tmet tmf rfietbch W bh tmf sft c` bhtfjfrs _ nf`bhfn gy

xWy xy < y x " x, y _ bs eh fqubveifhaf rfietbch.

U.> Ift H gf tmf sft c eii heturei hulgfrs ehn W gf tmf rfietbch ch H H nf`bhfn gy

(b) (e, g) W (a, n) en < ga(bb) (e, g) W (a, n) en (g + a) < ga (e + n)

prcvf tmet W. bs eh fqubveifhaf rfietbch bh feam aesf.

U.6 B` ? ] R ehn E, G ], tmfh prcvf tmet(b) `(E G) < (E) (G)(b) `(E G) (E) (G)

U.0 B ? ] R ehn E, G R, tmfh prcvf tmet

(b) ` 5 (E G) < ` 5 (E) 5 (G)(bb) ` 5 (E G) < ` 5 (E) 5 (G)(bbb) ` 5 (E G) < 5 (E) 5 (G)

U.: [rcvf tmet chiy chf-chf chtc uhatbch mes bhvfrsf uhatbch.

U.= [rcvf tmet tmf prcnuat c` ehy uhatbch wbtm tmf bnfhtbty uhatbch bs tmf uhatbch btsfi`.

U.2 [rcvf tmet tmf prcnuat c` ehy bhvfrtbgif uhatbch wbtm bts bhvfrsf 5 bs eh bnfhtbty uhatbch.

U.5; [rcvf tmet aclpcsbtf c` `uhatbchs bs esscabetbvf.

U.55 Ift ? E G ehn j ? G E suam tmet jc bs eh bnfhtbty uhatbch ch E ehn cj bs eh bnfhtbty uhatbch ch G.^mfh, j < 5

U.57 Ift ? E G ehn j ? G A gf chf-chf chtc uhatbchs. mfh jc` bs eisc chf-chf chtc ehn (jc`) 5 < 5cj 5

VHQCIPFN [WCGIFLQF]FWABQF B

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U.51 Ift ` ? H H gf nf`bhfn gy

`(x) <

-+

fvfhbshb` ,5hcnnbshb` ,5h

smcw tmet bs e gbofatbch.

U.5> Qmcw tmet tmf uhatbch

` ? W {1} W {5} jbvfh gy

`(x) . (b) Bs tmf gbhery cpfretbch *, nf`bhfn ch tmf sft H, jbvfh gy e * g < 7ge +

`cr eii e, g U, acllutetbvf3(bb) Bs tmf egcvf gbhery cpfretbch * esscabetbvf 3 (A.G.Q.F. 7;;=)

U6. Ift * gf e gbhery cpfretbch nf`bhfn gy e * g < 1e + >g 7. @bhn > * 6. (A.G.Q.F. 7;;=)

U0. B` (x) bs eh bhvfrtbgif uhatbch, bhn tmf bhvfrsf c` (x) . (A.G.Q.F. 7;;=)U=. B` (x) < x + : ehn j(x) < x :, bhn (`cj) (:). (A.G.Q.F. 7;;=)

U2. Ift ` ? H H gf nf`bhfn gy (h) .

(A.G.Q.F. 7;;2)U55. Qmcw tmet tmf rfietbch Q nf`bhfn ch tmf sft H H gy

(e, g) Q (a, n) e + n < g + a bs eh fqubveifhaf rfietbch. (A.G.Q.F. 7;5;)

U57. B` ? W W gf nf`bhfn gy `(x) < (1 x1

)5/1

, tmfh bhn c`(x). (A.G.Q.F. 7;5;)

U51. E gbhery cpfretbch * ch tmf sft {;, 5, 7, 1, >, 6} bs nf`bhfn es ? e * g <

+-+9++

0geb` ,0ge0geb` ,ge

Qmcw tmet zfrc bs tmf bnfhtbty cr tmbs cpfretbch ehn feam fiflfht e c` tmf sft bs bhvfrtbgif wbtm 0 e, gfbhj tmf bhvfrsf c` e. (A.G.Q.F. 7;55)

U5>. Ift ? W W gf nf`bhfn es (x) < 5;x + :. @bhn tmf uhatbch j ? W W suam tmet jc < `cj < BW .(A.G.Q.F. 7;55)

U56. Qmcw tmet ` ? H H, jbvfh gy

-

+, \ jbvfh gy `(x) < x 7 + >. Qmcw tmet ` bs bhvfrtbgif wbtmtmf bhvfrsf ` 5 c` ` jbvfh gy ` 5 (y) < >-y , wmfrf W + bs tmf sft c` eii hch-

hfjetbvf rfei hulgfrs. (AGQF 7;51)

GCEWN [WCGIFQF]FWABQF BB

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WFIE BCHQ EHN @VHA^BCHQ75

F]FWBABQF 5 (VHQCIPFN [WCGIFLQ)

5:.75

icj 5;

-+

x5x5

7;. (b) (5, ;) (bb)

-

eg

,e5

F]FWBABQF 7 (GCEWN [WCGIFLQ)

>. (b) yfs (bb) yfs 6. 1; 0. ` 5 (x) . j(x)