MAT 202 Introduction to Analysis [Pengantar Analisis]

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1 UNIVERSITI SAINS MALAYSIA Secod Semester Examiatio 2015/2016 Academic Sessio Jue 2016 MAT 202 Itroductio to Aalysis [Pegatar Aalisis] Duratio : 3 hours [Masa : 3 jam] Please check that this examiatio paper cosists of FIVE pages of prited material before you begi the examiatio [Sila pastika bahawa kertas peperiksaa ii megadugi LIMA muka surat yag bercetak sebelum ada memulaka peperiksaa ii] Istructios: Aswer THREE (3) questios [Araha: Jawab TIGA (3) soala] I the evet of ay discrepacies, the Eglish versio shall be used [Sekiraya terdapat sebarag percaggaha pada soala peperiksaa, versi Bahasa Iggeris hedaklah digua pakai] 2/-

2 - 2-1 (a) Let S be a subset of Defie what it meas for S to be bouded above (b) Let S be a subset of Defie what it meas for to be the least upper boud of S State the ompleteess Axiom for the set real umbers Does the set of all ratioal umbers satisfy the ompleteess Axiom? Justify your aswer by givig a example (e) Let f : X Y be a fuctio, with XY, ad A, B X Show that f ( A B) f ( A) f ( B) (f) Defie a coutable set Next determie whether the set { r : r 2} is coutable (g) Let A be a ifiite set Show that A has a coutably ifiite subset 1 (a) Biarka S suatu subset pada Takrifka apa yag dimaksudka dega S dibatasi dari atas (b) Biarka S suatu subset pada Takrifka apa yag dimaksudka dega sebagai batas atas terkecil bagi S Nyataka Aksiom Kelegkapa bagi set semua ombor yata Adakah set semua ombor isbah memeuhi Aksiom Kelegkapa? Berika jawapa ada dega meujukka cotoh (e) Biarka f : X Y suatu fugsi, dega XY, da A, B X Tujukka bahawa f ( A B) f ( A) f ( B) (f) Takrifka set terbilagka Seterusya, tetuka sama ada set { r : r 2} adalah terbilagka (g) Biarka A suatu set tidak terhigga Tujukka bahawa A mempuyai satu subset yag terbilagka secara tak terhigga 3/-

3 - 3-2 (a) Let a be a sequece Give the defiitio for the sequece coverge to a umber a Next show that a is uique (b) Let a be a coverget sequece a to (i) Show that a is bouded Is the coverse true? If yes, prove it ad if o, give a couter example (ii) Show that a is auchy Is the coverse true? If yes, prove it ad if o, give a couter example Let A be a subset of Defie the followig termiology: (i) Iterior poit of A (ii) Limit poit of A (iii) Isolated poit of A (iv) Boudary poit of A Fid for each type of poit i for the set A [ 2,19) (e) Determie whether the set A [ 2,19) is closed or ope (f) Let A be a subset of If x is a limit poit of A, the show that for each 0, the eighbourhood of x, that is, N x; cotais ifiitely may elemets of A Next, what ca you coclude about the set of limit poits of A if A is fiite? a suatu jujuka Berika takrif utuk jujuka a meumpu 2 (a) Biarka kepada ombor a Seterusya, tujukka a adalah uik (b) Biarka a suatu jujuka yag meumpu (i) Tujukka bahawa a adalah terbatas Adakah akasya bear? Jika ya, buktika da jika tidak, berika satu cotoh peyagkal (ii) Tujukka bahawa a adalah auchy Adakah akasya bear? Jika ya, buktika da jika tidak, berika satu cotoh peyagkal 4/-

4 - 4 - Biarka A suatu subset pada Takrifka istilah yag berikut: (i) Titik pedalama bagi A (ii) Titik had bagi A (iii) Titik terpecil bagi A (iv) Titik sempada bagi A ari setiap jeis titik di bahagia utuk set A [ 2,19) (e) Tetuka sama ada set A [ 2,19) adalah tertutup atau terbuka (f) Biarka A suatu subset pada Jika x adalah titik had bagi A, maka tujukka bahawa utuk setiap 0, jiraa bagi x, iaitu N x;, megadugi tak terhigga bayakya usur A Seterusya, apa yag dapat dirumuska jika set A adalah terhigga? 3 (a) State the Heie-Borel Theorem for Give a example of a compact set (b) Let A ad B be two compact sets Show that A B is compact Let f be a cotiuous real-valued fuctio o A ad g be a cotiuous fuctio o B, where f (A) B From the defiitio of cotiuity, prove that go f is cotiuous o A State the defiitio for the uiform cotiuity of a real-valued fuctio Let f x x Usig the defiitio, show that f is uiformly cotiuous o 2 the iterval [0,2] f defied by f x = x, ad x 0, 1 f coverges uiformly o 0, a for ay umber 0 a 1 f does ot coverge uiformly o 0, 1 (e) osider the sequece Show that but 5/-

5 - 5-3 (a) Nyataka Teorem Heie-Borel bagi Berika satu cotoh set padat (b) Biarka A, da B dua set padat Tujukka bahawa A B adalah padat Biarka f suatu fugsi yata pada A da g adalah fugsi selajar pada B, yag maa f (A) B Dega megguaka takrif utuk keselajara, buktika bahawa go f adalah selajar pada A Nyataka takrif utuk keselajara secara seragam bagi suatu fugsi 2 berilai yata Biarka f x x Dega megguaka takrifa, tujukka bahawa f adalah selajar secara seragam pada selag [0,2] (e) Pertimbagka jujuka f yag ditakrifka oleh =, da x 0, 1 Tujukka bahawa 0, a utuk sebarag ombor 0 a 1 tetapi seragam pada 0, 1 f x x f meumpu secara seraga pada f tak meumpu secara - ooo 0 ooo -

MSG388 Mathematical Algorithms for Computer Graphics [Algoritma Matematik untuk Grafik Komputer]

UNIVERSITI SAINS MALAYSIA Secod Semester Examiatio 015/016 Academic Sessio Jue 016 MSG388 Mathematical Algorithms for Computer Graphics [Algoritma Matematik utuk Grafik Komputer] Duratio : 3 hours [Masa