Std. 9th, Algebra, Marathi Medium, Maharashtra Board · No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic,

No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher.

yeerpeieefCele F³eÊee veJeJeer

Printed at: Repro India Ltd., Mumbai

Fourth Edition: April 2016

þUkeÀ Jewefμe<ìîes: • veJeerve DeY³eeme¬eÀceeJej DeeOeeefjle. • mebHetCe& DeY³eeme¬eÀcee®ee HeefjHetCe& Dee{eJee. • Òel³eskeÀ Heeþe®³ee meg©Jeeleeruee Òeμvemeb®eeb®eer Je mejeJee®³ee Òeμveeb®eer Òeμve ÒekeÀejebvegmeej efJeYeeieCeer. • HeeþîeHegmlekeÀeleerue meJe& mJeeO³ee³e Je Òeμvemeb®eeb®eer GÊejs meceeefJe<ì. • mejeJeemeeþer DeefOekeÀ Òeμve. • meKeesue DeY³eemeemeeþer yengHe³ee&³eer Òeμve meceeefJe<ì. • ÒeμveHes{erleerue Òeμveeb®eer meJe&meceeJesμekeÀ GkeÀue.

P.O. No. 16147

cenejeä^ jep³e ceeO³eefcekeÀ Je G®®e ceeO³eefcekeÀ efμe#eCe ceb[U, HegCes ³eeb®³eeÜejs efve³eesefpele DeY³eeme¬eÀceeJej DeeOeeefjle

10041_10550_JUP

ÒemleeJevee vecemkeÀej efJeÐeeLeea efce$e-cewef$eCeeRvees, F³eÊee veJeJeerle HeoeHe&Ce kesÀuesu³ee meJe& efJeÐeeLeea efce$ecewef$eCeeR®es ceve:HetJe&keÀ mJeeiele. ieefCele ne efJe<e³e μeeues³e DeY³eeme¬eÀcee®ee SkeÀ DeefJeYeep³e IeìkeÀ Deens. yeerpeieefCele ne ieefCelee®ee keÀCee Deens. owvebefove peerJeveele HeeJeueesHeeJeueer yeerpeieefCeleeμeer mebyebOe ³esle Demelees. ne efJe<e³e efμekeÀu³eeJej leke&Àμeg× efJe®eej, ceeb[Ceer, DeekeÀ[scees[erle De®etkeÀlee, efJeμues<eCeelcekeÀ efJe®eej Deeoer iees<ìeRceO³es efJeÐeeLeea Heejbiele nesleele. ³ee efJe<e³eeceO³es yewefpekeÀ jeμeer, oesve ®eueebleerue js<eer³e meceerkeÀjCes lemes®e keWÀêer³e ÒeJe=Êeer®eer HeefjceeCes, meebefK³ekeÀ meeceûeer®es JeieeakeÀjCe Fl³eeoer mebkeÀuHeveeb®ee meceeJesμe neslees. yeerpeieefCele efJe<e³ee®ee DeY³eeme meesHee JneJee lemes®e cegueeb®ee DeelceefJeμJeeme Jee{eJee ³eemeeþer Deecner `yeerpeieefCele : F³eÊee veJeJeer' ns ³ee efJe<e³eeJej meKeesue Je mebHetCe& ceeie&oμe&ve keÀjCeejs HegmlekeÀ DeeHeu³eemeceesj meeoj keÀjle Deenesle. Òeμve Je GÊejeb®³ee mJeªHeeleerue ns HegmlekeÀ efJeÐeeL³ee¥vee Òel³eskeÀ mebkeÀuHevee HetCe&HeCes mecepeC³eele meene³³e keÀjsue. Òel³eskeÀ ÒekeÀjCee®³ee meg©Jeeleeruee Heeþeleerue meJe& mJeeO³ee³eeb®es Je Òeμvemeb®eeb®es Òeμve ÒekeÀejebJej DeeOeeefjle efJeYeepeve efoues Deens; p³eecegUs efJeÐeeL³ee¥vee efJeefJeOe Òeμve ÒekeÀej mecepeC³eeme ceole nesF&ue. ³ee HegmlekeÀele Heeþeleerue mJeeO³ee³e, Òeμvemeb®e Je ÒeμveHes{er®³ee GÊejeb®ee meceeJesμe kesÀuesuee Deens. l³ee®eyejesyej mejeJeemeeþer DeefOekeÀ Òeμve DeeefCe yengHe³ee&³eer Òeμvener meceeefJe<ì kesÀuesues Deensle. meJe& DeeuesKe meg³eesi³e ÒeceeCeevegmeej Deensle. ³ee HegmlekeÀe®es DeeCeKeer SkeÀ Jewefμe<ìîe cnCepes ³ee®eer DeekeÀ<e&keÀ ceeb[Ceer. ³ee HegmlekeÀeÜejs F³eÊee oneJeer®ee Hee³ee HekeÌkeÀe keÀjC³eemeeþer, lemes®e DeefOekeÀeDeefOekeÀ mejeJeemeeþer efJeÐeeLeea vekeÌkeÀer®e ÒeJe=Êe nesleerue Deμeer Deecnebuee Kee$eer Deens. efpeÎerves, ef®ekeÀeìerves DeY³eemeeuee ueeiee, ³eμe legce®es®e Deens! HegmlekeÀeefJe<e³eer DeefOekeÀ met®evee Je DeefYeÒee³e ³eeb®eer Deecner Jeeì yeIelees³e. Oev³eJeeo!

Hejer#esmeeþer neefo&keÀ μegYes®íe ! ÒekeÀeμekeÀ

¬eÀceebkeÀ ÒekeÀjCes He=ÿ ¬eÀ.

1 meb®e 1

2 JeemleJe mebK³ee 22

3 yewefpekeÀ jeμeer 65

4 oesve ®eueebleerue js<eer³e meceerkeÀjCes 103

5 DeeuesKe 138

6 iegCeesÊej Je ÒeceeCe 201

7 meebefK³ekeÀer 233

8. ÒeμveHes{er 269

Deveg¬eÀceefCekeÀe

1

ÒekeÀjCe 01: meb®e

Òeμve ÒekeÀej DeY³eeme Òeμve ¬eÀceebkeÀ

meb®ee®eer J³eeK³ee 1.1 Òe.1Òeμvemeb®e -1 Òe.1

meb®e efueefnC³ee®³ee He×leer

1.1 Òe.2, 3, 4

mejeJeemeeþer Òeμve(DeY³eeme 1.1 Jej DeeOeeefjle)

Òe.1, 2

Òeμvemeb®e -1 Òe.2, 3, 10

meb®eeb®es ÒekeÀej

1.2 Òe.1, 2, 3, 4

mejeJeemeeþer Òeμve(DeY³eeme 1.2 Jej DeeOeeefjle) Òe.1, 2


GHemeb®e DeeefCe efJeμJemeb®e

1.3 Òe.1, 2, 4, 5

mejeJeemeeþer Òeμve(DeY³eeme 1.3 Jej DeeOeeefjle) Òe.1, 3, 4


meb®eebJejerue ef¬eÀ³ee DeeefCe l³eeb®es iegCeOece&

1.4 Òe.1, 2, 3, 4, 5 mejeJeemeeþer Òeμve

(DeY³eeme 1.4 Jej DeeOeeefjle) Òe.1, 2, 3, 4

Òeμvemeb®e -1 Òe.6, 7, 8, 13, 14, 21, 23

meb®eeleerue IeìkeÀeb®eer mebK³ee

1.5 Òe.1, 2, 5

mejeJeemeeþer Òeμve(DeY³eeme 1.5 Jej DeeOeeefjle) Òe.2

Òeμvemeb®e -1 Òe.15, 18

meb®eeJejerue μeeefyokeÀ GoenjCes

1.5 Òe. 3, 4

mejeJeemeeþer Òeμve(DeY³eeme 1.5 Jej DeeOeeefjle) Òe.3, 4


Jesve ef®e$e keÀe{e

1.3 Òe.3mejeJeemeeþer Òeμve

(DeY³eeme 1.3 Jej DeeOeeefjle) Òe.2

mejeJeemeeþer Òeμve(DeY³eeme 1.5 Jej DeeOeeefjle)

Òe.1, 5

Òeμvemeb®e -1 Òe.20

meb®e 01

2

F. 9 Jeer : yeerpeieefCele

ÒemleeJevee Keeueerue GoenjCes efJe®eejele I³ee: i. ûebLeeue³eeleerue HegmlekeÀeb®ee mebûen. ii. ogkeÀeveeleerue keÀHe[îeeb®ee mebûen. ³ee GoenjCeebleerue Jemlet þUkeÀHeCes efomeleele. ns megmHe<ì mecetn Demeleele. Keeueerue GoenjCes efJe®eejele I³ee: i. Jeiee&leerue ngμeej efJeÐeeLeea. ii. μenjeleerue Deeveboer J³ekeÌleer. ’ngμeej “ Je ’Deeveboer“ ³ee yeeyeer meeHes#e Deensle. SkeÀe J³ekeÌleer®³ee celes p³ee J³ekeÌleer ngμeej efkebÀJee Deeveboer Deensle l³ee J³ekeÌleer ogmeN³ee®³ee celes ngμeej Je Deeveboer Demeleerue®e Demes veener. efouesuee mecetn ne megmHe<ì mecetn Deens efkebÀJee veener ns þjefJeCes cenÊJee®es Demeles. Jemletb®³ee megmHe<ì mecetneuee `meb®e' Demes cnCeleele. pece&ve ieefCele%e pee@pe& keBÀìj (1845-1918) ns `meb®e efme×eblee®es' pevekeÀ cnCetve DeesUKeues peeleele. `meb®e efme×eble' ner ieefCelee®³ee DeY³eemeeleerue cetueYetle mebkeÀuHevee Deens. 1.1 meb®eeb®eer J³eeK³ee meb®e: Jemletb®³ee megmHeä mecetneuee `meb®e' Demes cnCeleele. Goe:

i. efJe<ece vewmeefie&keÀ mebK³eeb®ee mecetn ii. HetCe& mebK³eeb®ee mecetn meb®eeleerue Òel³eskeÀ Jemlet ner l³ee meb®ee®ee ÌeìkeÀ' Deens Demes cnCeleele. Goe: i. efJe<ece vewmeefie&keÀ mebK³ee Demeuesu³ee meb®eele 1, 3, 5, 7, … ns

IeìkeÀ Deensle. ii. HetCe& mebK³ee Demeuesu³ee meb®eele 0, 1, 2, 3, … ns IeìkeÀ

Deensle. megmHe<ì vemeuesu³ee IeìkeÀeb®³ee mecetneves meb®e le³eej nesT μekeÀle veener. Deμee meb®eeble meJe&meeOeejCeHeCes meesHes, ®eebieues, DeeJe[les Fl³eeoer meeHes#e yeeyeer meceeefJe<ì Demeleele. Goe: ûebLeeue³eeleerue ®eebieu³ee HegmlekeÀeb®ee mecetn. ³esLes `®eebieu³ee' ner meeHes#e yeeye Demetve l³ee®ee DeLe& Òel³eskeÀ J³ekeÌleervegmeej yeouet μekeÀlees.

ue#eele þsJeC³ee®es cenÊJee®es cegÎs: 1. meb®e oμe&efJeC³eemeeþer A, B, C…….. X, Y, Z,

ner Fbûepeerleerue Heefnu³ee efueHeerleerue De#ejs JeeHejleele lej meb®eeleerue IeìkeÀ oeKeefJeC³eemeeþer meeceev³eHeCes a, b, c …….x, y, z ner Fbûepeerleerue ogmeN³ee efueHeerleerue De#ejs JeeHejleele.

2. pej ‘r’ ne meb®e P ®ee IeìkeÀ Demesue lej r P Demes efueefnues peeles DeeefCe les Heg{erueÒeceeCes Jee®eues peeles.

i. ‘r’ ne P meb®eeμeer efveie[erle Deens. efkebÀJee ii. ‘r’ ne P meb®ee®ee Yeeie Deens. efkebÀJee iii. ‘r’ ne P meb®ee®ee IeìkeÀ Deens. ³eeceO³es ‘’ ³ee ef®evne®ee DeLe& èfveie[erle DemeCes' efkeÀbJee

`Yeeie DemeCes' efkebÀJee ÌeìkeÀ DemeCes' Demee neslees. 3. pej ‘r’ ne P meb®ee®ee IeìkeÀ vemesue lej r P Demes

efueefnues peeles DeeefCe les Heg{erueÒeceeCes Jee®eues peeles. i. ‘r’ ne P meb®eeμeer efveie[erle veener efkebÀJee ii. ‘r’ ne P meb®ee®ee Yeeie veener efkebÀJee iii. ‘r’ ne P meb®ee®ee IeìkeÀ veener. ³eeceO³es ‘’ ³ee ef®evne®ee DeLe& èfveieef[le vemeCes' efkebÀJee `®ee

Yeeie vemeCes' efkebÀJee `®ee IeìkeÀ vemeCes' Demee neslees. 4. vewmeefie&keÀ mebK³ee meb®e, HetCe& mebK³ee meb®e, HetCee¥keÀ mebK³ee meb®e,

Heefjces³e mebK³ee meb®e, JeemleJe mebK³ee meb®e ns Deveg¬eÀces N,

W, I, Q, R ³ee De#ejebveer oμe&Jeleele. 1.2 meb®e efueefnC³ee®³ee He×leer meb®e efueefnC³ee®³ee oesve He×leer Deensle: a. ³eeoer He×leer (Listing Method efkebÀJee Roster form)

b. iegCeOece& He×leerr (Rule Method efkebÀJee Set builder form) a. ³eeoer He×leer (Listing method efkebÀJee Roster form) ³ee He×leerle: i. meb®ee®es IeìkeÀ ceefnjHeer kebÀmeele efueefnues peeleele. ii. Òel³eskeÀ IeìkeÀ ne HeÀkeÌle SkeÀoe®e efueefnuee peelees. iii. Òel³eskeÀ IeìkeÀ ne mJeuHeefJejece osTve JesieUe kesÀuee

peelees. iv. ³eeceO³es IeìkeÀeb®ee ¬eÀce cenÊJee®ee vemelees. Goe. A = {a, b, c, d, e} efkebÀJee A = {b, d, a, c, e} ns meceeve meb®e Deensle pes

Heefnueer Hee®e Fbûepeer De#ejs oμe&efJeleele.

3


³eeoer He×leerves meb®e efueefnC³ee®eer keÀener GoenjCes Heg{erue ÒeceeCes Deensle:

i. L ne “fatal” ³ee μeyoeleerue De#ejeb®ee meb®e Deens. L = {f, a, t, l} ii. M ne 5 Hes#ee keÀceer HetCee¥keÀ mebK³eeb®ee meb®e Deens. M = {… , 3, 2, 1, 0, 1, 2, 3, 4} iii. O ne 1 les 100 He³e¥le®³ee meJe& vewmeefie&keÀ mece

mebK³eeb®ee meb®e Deens. O = {2, 4, 6, 8, … , 100} b. iegCeOece& He×leer (Rule method efkebÀJee Set builder

form) meb®e efueefnC³ee®³ee iegCeOece& He×leerceO³es meb®eeleerue IeìkeÀeb®es

JeCe&ve kesÀues peeles. l³ee JeCe&veecegUs efkebÀJee efJeefμe<ì efve³ece mHe<ì kesÀu³eecegUs meb®eeleerue IeìkeÀeb®es SkeÀcesJelJe efveefμ®ele kesÀues peeles.

Goe: i. Y = {x|x ne Fbûepeer JeCe&ceeuesleerue mJej Deens.}

Jejerue uesKeve He×leerle ceefnjHeer kebÀme `meb®eekeÀefjlee' DeeefCe GYeer jsIe (|) ns `Demes keÀer' ³ee μeyoekeÀefjlee JeeHejleele.

Y ne meb®e Heg{erue ÒeceeCes Jee®euee peelees: ’Y ne Deμee ‘x’ IeìkeÀeb®ee meb®e Deens keÀer ‘x’ ne

Fbûepeer JeCe&ceeuesleerue mJej Deens.“ ii. B = {x|x W, x < 10} B ne meb®e Heg{erue ÒeceeCes Jee®euee peelees: ’B ne Deμee ‘x’ IeìkeÀeb®ee meb®e Deens keÀer, x ner

10 Hes#ee ueneve Demeuesueer HetCe& mebK³ee Deens.“

ìerHe: keÀener JesUsme ‘|’ (GYeer jsIe) ³eeSsJepeer ‘:’ (efJemeie&) JeeHejuee peelees. DeY³eeme 1.1 1. KeeueerueHewkeÀer keÀesCeles mecetn meb®e Deensle ? i. cetU mebK³eeb®ee mecetn. ii. ³ee ÒekeÀjCeeleerue meesH³ee GoenjCeeb®ee mecetn. iii. legce®³ee μeeUsleerue ®eebieu³ee efμe#ekeÀeb®ee mecetn. iv. legce®³ee Jeiee&leerue cegueeR®ee mecetn. v. efJe<ece vewmeefie&keÀ mebK³eeb®ee mecetn. GkeÀue: i. ne meb®e Deens. ii. `meesH³ee GoenjCeeb®ee' ³ee®ee DeLe& J³ekeÌleerHejlJes yeouelees

keÀejCe ner meeHes#e yeeye Deens. l³eecegUs ne meb®e veener.

iii. ®eebieu³ee efμe#ekeÀeb®eer efveJe[ Òel³eskeÀ efJeÐeeL³ee&vegmeej yeoueles keÀejCe ®eebieues ner meeHes#e yeeye Deens. l³eecegUs ne meb®e veener.

iv. ne meb®e Deens. v. ne meb®e Deens. 2. Keeueerue meb®e ³eeoer He×leerves efuene.

i. A = {x|x ne ûesiejer³eve Je<ee&leerue 30 efoJemeeb®ee ceefnvee veener }

ii. B = {y|y ne FbêOeveg<³eeleerue jbie Deens}

iii. C = {x|x ne HetCee¥keÀ Demetve 4 < x < 4}

iv. D = {x|x I, 3 < x 3} v. E = {x|x = (n 1)3, n < 3, n W}

GkeÀue: i. A = {peevesJeejer, HesÀye´gJeejer, cee®e&, ces, peguew, Dee@iemì,

Dee@keÌìesyej,ef[meWyej}

ii. B = {peebYeUe, HeejJee, efveUe, efnjJee, efHeJeUe, veeefjbieer, leebye[e,}

iii. C = {3, 2, 1, 0, 1, 2, 3} iv. D = {2, 1, 0, 1, 2, 3} v. n = 0, 1, 2, ¿ee efkeÀceleer Ieeuetve E = {1, 0, 1} 3. Keeueerue meb®e iegCeOece& He×leerves efuene. i. F = {5, 10, 15, 20} ii. G = {9, 16, 25, 36, … , 81} iii. H = {5, 52, 53, 54} iv. X = {8, 8}

v. Y =

1 1 1 11, , , ,

8 27 64 125

GkeÀue: i. F = {x|x = 5n, n N, n 4} ii. G = {x|x = n2, n N, 3 n < 10} iii. H = {x|x = 5n, n N, n 4} iv. X = {x| x ®ee Jeie& 64 Deens}

efkebÀJee X = {x|x ns 64 ®es Jeie&cetU Deens}

v. Y = {x|x = 3

1

n, n N, n 5}

4. p³eeb®ee Jeie& efJe<ece mebK³ee ³esF&ue Deμee Heefnu³ee Hee®e

Oeve mebK³eeb®ee meb®e efuene. GkeÀue: P = {1, 3, 5, 7, 9}

4

F. 9 Jeer : yeerpeieefCele 1.3 Jesve ef®e$es ³eguej ³ee Leesj ieefCele%eeves meb®eeb®ee DeY³eeme keÀjC³eemeeþer Deeke=Àleer efkebÀJee ef®e$ee®ee JeeHej keÀjC³ee®eer keÀuHevee ceeb[ueer. l³eeveblej, efye´ìerμe leke&Àμeem$e%e pee@ve Jesve (1834-1923) ³eebveer meb®eeb®ee DeY³eeme keÀjC³eemeeþer ³ee mebkeÀuHeves®ee GHe³eesie Je efJekeÀeme kesÀuee. meb®e oμe&efJeC³eemeeþer JeeHeju³ee peeCeeN³ee Deeke=Àl³eebvee `Jesve ef®e$es'Demes cnCeleele. Jesve ef®e$eebceO³es, meb®e ne yebefomle Deeke=Àleerves oμe&efJeuee peelees Je l³ee meb®ee®es IeìkeÀ l³ee yebefomle Deeke=Àleerle efyebotb®³ee meene³³eeves oμe&efJeues peeleele. Jesve ef®e$es oeKeefJeC³eemeeþer Dee³ele, Jeleg&U, ef$ekeÀesCe, F. Deeke=Àl³eeb®ee JeeHej kesÀuee peelees. Goe. 1.4 meb®eeb®es ÒekeÀej i. SkeÀ IeìkeÀ meb®e : p³ee meb®eeceO³es HeÀkeÌle SkeÀ®e IeìkeÀ

Demelees l³eeme SkeÀ IeìkeÀ meb®e cnCeleele. Goe. a. A = {5} b. B = {x|x + 3 = 0} meb®e B ceO³es HeÀkeÌle SkeÀ®e IeìkeÀ Deens lees cnCepes 3 ii. efjkeÌle meb®e : p³ee meb®eeceO³es SkeÀner IeìkeÀ vemelees l³eeme

efjkeÌle meb®e cnCeleele. ne meb®e {} efkebÀJee (phi) ³ee ef®evneves oμe&efJeleele.

Goe. a. A = {a|a ner vewmeefie&keÀ mebK³ee Deens, 5 < a < 6}

A = { } efkebÀJee A = b. B = {x|x ner vewmeefie&keÀ mebK³ee Deens, x < 1}

B = iii. meeble meb®e : p³ee meb®eele IeìkeÀeb®eer ceespeCeer efveefμ®ele

efþkeÀeCeer Kebef[le nesles l³eeme meeble meb®e cnCeleele. Goe: A = {1, 2, 3, 4, 5, 6, 7}

B = {x|x ns DeeþJe[îeeleerue Jeej Deensle}

Jejerue A DeeefCe B meb®eele meeble IeìkeÀ Deensle. meb®e A DeeefCe B ns meeble meb®e Deensle. iv. Deveble meb®e : p³ee meb®eele IeìkeÀeb®eer ceespeCeer keÀesþsner

Kebef[le nesle veener l³eeme Deveble meb®e cnCeleele. Goe: P = {1 , 2, 3, 4, 5, 6, …} W = {x|x ner HetCe& mebK³ee Deens}

Jejerue meb®e P DeeefCe W ceOeerue IeìkeÀeb®eer ceespeCeer keÀjlee ³esT μekeÀle veener Je leer Kebef[lener nesle veener. cnCetve P Je W ns Deveble meb®e Deensle.

ìerHe: i. X = {0} ne efjkeÌle meb®e veener keÀejCe ‘0’ ne X

meb®ee®ee IeìkeÀ Deens. ii. efjkeÌle meb®e ne meeble meb®e Demelees. iii. vewmeefie&keÀ mebK³ee, HetCe& mebK³ee, HetCee¥keÀ mebK³ee,

Heefjces³e mebK³ee DeeefCe JeemleJe mebK³ee ns meJe& Deveble meb®e Deensle.

DeY³eeme 1.2 1. KeeueerueHewkeÀer keÀesCeles meb®e `SkeÀ IeìkeÀ meb®e' Deensle? i. A = 16x x

ii. B = {y|y2 = 36} iii. C = {p|p I, p3 = 8} iv. D = {q|(q 4)2 = 0} v. E = {x|1 + 2x = 3x, x W} GkeÀue: i. x = 16 x = 256 A = {256} ne SkeÀ IeìkeÀ meb®e Deens. ii. y2 = 36 y = 6 B = {–6, +6} ne SkeÀ IeìkeÀ meb®e veener. iii. p3 = 8 p3 = (2)3 p = 2 C = {2} ne SkeÀ IeìkeÀ meb®e Deens.

A . a . e . i

. o . u

. a

. b . c

. d . e

C

. 0

. 2

. 1

. 2

. 4

. 6

. 8

B

D

5


iv. (q 4)2 = 0 q 4 = 0 q = 4 D = {4} ne SkeÀ IeìkeÀ meb®e Deens. v. 1 + 2x = 3x 1 = 3x 2x 1 = x x = 1 E = {1} ne SkeÀ IeìkeÀ meb®e Deens. 2. KeeueerueHewkeÀer keÀesCeles meb®e èfjkeÌle meb®e' Deensle ? i. A ne meJe& mece cetU mebbK³eeb®ee meb®e Deens.

ii. B = {x|x ner Yeejlee®eer jepeOeeveer Deens}

iii. F = {y|y ne oesve meceeblej js<eeb®ee ísoveefyebot Deens}

iv. G = {z|z N, 3 < z < 4} v. H = {t|t ne ®eej yeepet Demeuesuee ef$ekeÀesCe Deens}

GkeÀue: i. A = {2} ne efjkeÌle meb®e veener. ii. B = { efouueer }

ne efjkeÌle meb®e veener. iii. meceeblej js<ee SkeÀceskeÀebvee ísole veener. F = { } ne efjkeÌle meb®e Deens. iv. z ner vewmeefie&keÀ mebK³ee Deens. 3 Je 4 ®³ee ceO³es SkeÀner

vewmeefie&keÀ mebK³ee veener. G = { } ne efjkeÌle meb®e Deens. v. ef$ekeÀesCe ner leerve yeepet Demeuesueer Deeke=Àleer Demeles. H = { } ne efjkeÌle meb®e Deens. 3. Keeueerue meb®eeb®es ‘meeble’ efkebÀJee ‘Deveble’ meb®eeceO³es

JeieeakeÀjCe keÀje. i. A = {1, 3, 5, 7, …} ii. B = {101, 102, 103, … , 1000}

iii. C = {x|x Q, 3 < x < 5} iv. D = {y|y = 3n, n N} GkeÀue: i. ³esLes, IeìkeÀeb®eer ceespeCeer keÀesþsner Kebef[le nesle veener. A ne Deveble meb®e Deens. ii. ³esLes, IeìkeÀeb®eer ceespeCeer 1000 ³ee mebK³esJej Kebef[le nesles. B ne meeble meb®e Deens. iii. 3 Je 5 ³ee DebkeÀebceO³es Deveble Heefjces³e mebK³ee Deensle. C ne Deveble meb®e Deens. iv. ³esLes, IeìkeÀeb®eer ceespeCeer keÀesCel³eener ìHH³eeJej Kebef[le nesle

veener. D ne Deveble meb®e Deens. 4. mecepee G = {x|x ne legce®³ee Jeiee&leerue cegueiee Deens}

DeeefCe H = {y|y ner legce®³ee Jeiee&leerue cegueieer Deens}

lej G DeeefCe H ns keÀesCel³ee ÒekeÀej®es meb®e Deensle ?

GkeÀue: G DeeefCe H ns meeble meb®e Deensle. 1.5 GHemeb®e pej meb®e Y ceOeerue Òel³eskeÀ IeìkeÀ ne meb®e X ®ee IeìkeÀ Demesue, lej Y uee meb®e X ®ee GHemeb®e cnCeleele. lees Y X ³ee ef®evneves oμe&efJeuee peelees. pej ‘a’ ne IeìkeÀ Y meb®eeμeer mebyebefOele Demesue, lej lees X

meb®eeμeerner mebyebefOele Demelees. Hejbleg pej a Y DeeefCe a X lej Y ne X ®ee GHemeb®e vemelees efkebÀJee Y X.

Goe: pej Y = {b, z} DeeefCe X = {b, l, z} lej DeeHeCe Demes cnCelees keÀer Y X.

pej Y ne X ®ee GHemeb®e Demesue DeeefCe meb®e Y ceO³es vemeuesuee efkeÀceeve SkeÀ IeìkeÀ meb®e X ceO³es peemleer®ee Demesue, lej meb®e Y ne meb®e X ®ee ³egkeÌle GHemeb®e Demelees. lees Heg{erueÒeceeCes oμe&Jeleele. Y X. meb®e X uee meb®e Y ®ee `Deeflemeb®e' cnCeleele DeeefCe lees X Y Demee oμe&efJeleele. pej, X = {a, b} DeeefCe Y = {b, a}, lej meb®e X ne meb®e Y ®ee GHemeb®e Deens DeeefCe meb®e Y ne meg×e meb®e X ®ee GHemeb®e Deens. ³esLes meb®e X ne meb®e Y ®ee De³egkeÌle GHemeb®e Deens.

6


lees X Y Demee oμe&efJeleele DeeefCe l³ee®es Jee®eve ’X ne Y ®ee De³egkeÌle GHemeb®e Deens.“ Demes keÀjleele. lemes®e Y meg×e meb®e X ®ee De³egkeÌle GHemeb®e Deens. lees Y X Demee oμe&efJeleele DeeefCe l³ee®es Jee®eve ’Y ne X ®ee De³egkeÌle GHemeb®e Deens.“ Demes keÀjleele. ìerHe: i. Òel³eskeÀ meb®e ne mJele:®ee GHemeb®e Demelees. cnCepes®e

Y Y. ii. efjkeÌle meb®e ne Òel³eskeÀ meb®ee®ee GHemeb®e Demelees. cnCepes®e X. 1.6 efJeμJemeb®e efouesues meJe& meb®e, p³ee meb®ee®es Ghemeb®e nesleerue Deμee efveJe[uesu³ee Hejbleg efjkeÌle vemeuesu³ee meb®eeuee efouesu³ee meb®eeb®³ee meboYee&leerue èfJeμJemeb®e' cnCeleele. efJeμJemeb®e ne ‘U’ De#ejeves oμe&Jeleele. Goe: A = {x|x ner μeeUsleerue YeeweflekeÀμeem$e Òe³eesieμeeUe Deens.}

B = {y|y ner μeeUsleerue jmee³eveμeem$e Òe³eesieμeeUe Deens}

C = {z|z ner μeeUsleerue peerJeμeem$e Òe³eesieμeeUe Deens.}

U = {l|l ner μeeUsleerue Òe³eesieμeeUe Deens.}

³eeJeªve Demes efometve ³esles keÀer, A U, B U, C U.

meb®e U ne meb®e A, B DeeefCe C ®ee efJeμJemeb®e Deens. ìerHe: efJeμJemeb®e ne Demee meb®e Deens keÀer, pees SKeeÐee

GoenjCeemeeþer þjefJeuee Demelee keÀOeerner yeouele veener. Jesve Deeke=Àleerle efJeμJemeb®e ne meJe&meeOeejCeHeCes Dee³eleeves oμe&efJeuee peelees. DeY³eeme 1.3 1. Keeueerue meb®eeb®es efvejer#eCe keÀje Je l³eemebyebefOele

Òeμveeb®eer GÊejs efuene: A = cegbyeF&ceOeerue meJe& jefnJeeμeeb®ee meb®e B = YeesHeeUceOeerue meJe& jefnJeeμeeb®ee meb®e C = cenejeäêleerue meJe& jefnJeeμeeb®ee meb®e D = Yeejleeleerue meJe& jefnJeeμeeb®ee meb®e E = ceO³eÒeosμeeleerue meJe& jefnJeeμeeb®ee meb®e i. meb®e A DeeefCe meb®e C ceOeerue `GHemeb®e-mebyebOe'

efuene.

ii. meb®e E DeeefCe meb®e D ceOeerue `GHemeb®e-mebyebOe' efuene

iii. keÀesCelee meb®e Flej meb®eeb®³ee meboYee&le èfJeμJemeb®e' cnCetve efveJe[lee ³esT μekesÀue ?

GkeÀue: i. cegbyeF&leerue meJe& jefnJeemeer ns cenejeäê®es jefnJeemeer Deensle. A C ii. ceO³eÒeosμeeleerue meJe& jefnJeemeer ns Yeejlee®es jefnJeemeer Deensle. E D iii. cegbyeF&, cenejeä^, YeesHeeU, ceO³eÒeosμe ns Yeejlee®es IeìkeÀ

Deensle. D ne efJeμJemeb®e cnCetve efveJe[lee ³esT μekesÀue. 2. pej A = {a, b, c}, B = {a}, C = {a, b}, lej

i. keÀesCekeÀesCeles meb®e A ³ee meb®ee®es Gef®ele (³egkeÌle) GHemeb®e Deensle ?

ii. meb®e C ®ee Deeflemeb®e keÀesCelee Deens ?

GkeÀue: i. meb®e B Je meb®e C ®es IeìkeÀ ns meb®e A ®es IeìkeÀ Deensle.

lemes®e c ne Demee IeìkeÀ Deens pees meb®e B Je meb®e C ®ee IeìkeÀ veener Hejbleg lees meb®e A ceO³es Deens.

meb®e B DeeefCe meb®e C ns meb®e A ®es ³egkeÌle GHemeb®e Deensle.

ii. meb®e A ne meb®e C ®ee Deeflemeb®e Deens. cnCepes®e, A C.

3. Keeueerue meb®eebceOeerue `GHemeb®e-mebyebOe' Jesve

Deeke=Àleer®³ee meene³³eeves oeKeJee. A = {2, 4} B = {x|x = 2n, n < 5, n N} C = {x|x ner mece vewmeefie&keÀ mebK³ee Deens, x 16}

GkeÀue: A = {2, 4}, B = {2, 4, 8, 16} C = {2, 4, 6, 8, 10, 12, 14, 16} A B C

A B .12

.8

.6

.2

.4

.16

.14

.10

C

7


4. pej A B DeeefCe B C, lej A C ns efme× keÀje. (met®evee x A ves megªJeele keÀje Je x C ns oeKeJee) GkeÀue: mecepee, x A ....(i) Hejbleg, A B

x B

B C,

x C ....(ii) (i) Je (ii) Jeªve, A C 5. pej X = {1, 2, 3} Demesue lej, meb®e X ®es meJe& μekeÌ³e

GHemeb®e efuene. GkeÀue: meb®e X ®es meJe& μekeÌ³e GHemeb®e Heg{erueÒeceeCes : i. { } efkebÀJee ….[efjkeÌle meb®e ne Òel³eskeÀ meb®ee®ee GHemeb®e Demelees.] ii. {1} iii. {2} iv. {3} v. {1, 2} vi. {1, 3} vii. {2, 3} viii. {1, 2, 3} ….[Òel³eskeÀ meb®e ne mJele:®ee®e GHemeb®e Demelees.] 1.7 meb®eebJejerue ef¬eÀ³ee a. meceevelee: pej A ne B ®ee GHemeb®e Demesue Je B ne A ®ee GHemeb®e

Demesue lej A DeeefCe B ³eebvee meceeve meb®e Demes cnCeleele. les A = B ³eeves oμe&efJeues peeleele.

A DeeefCe B meb®eele lebleesleble les®e IeìkeÀ Demeleele. pej meb®e A Je B ®es IeìkeÀ meceeve vemeleerue lej les A B

Demes oμe&efJeleele. ìerHe: pej meb®e A DeeefCe B meceeve Deensle Demes efme× keÀje³e®es

Demesue lej l³eemeeþer A B DeeefCe B A ns efme× keÀjCes iejpes®es Deens.

i. mecepee, A = {x|x = 2n, n N DeeefCe x < 10}

DeeefCe B = {2, 4, 6, 8}

A = {2, 4, 6, 8} A B DeeefCe B A

A = B

ii. mecepee, P = {x|x ner efJe<ece vewmeefie&keÀ mebK³ee Deens, x < 8}

DeeefCe Q = {y|y ner mece vewmeefie&keÀ mebK³ee Deens, y <10}

³eeoer He×leerÒeceeCes P = {1, 3, 5, 7} Q = {2, 4, 6, 8} P Q DeeefCe Q P

P Q b. meb®eeb®ee íso: mecepee, A DeeefCe B

ns oesve meb®e efoues Deensle lej A DeeefCe B ceOeerue meeceeF&keÀ IeìkeÀeb®³ee meb®eeuee ísomeb®e Demes cnCeleele. lees A B

Demee oμe&efJeleele. l³ee®es Jee®eve ‘A íso

B’ Demes

keÀjleele. Goe: mecepee, A = {1, 3, 5, 7, 9} B = {3, 9, 12} A B = {3, 9} efkebÀJee A B = {x|x A DeeefCe x B} Jesve Deeke=Àleer®³ee íe³eebefkeÀle Yeeieeves meb®e A Je B ®ee

ísomeb®e oμe&efJeuee Deens. ísomeb®ee®es iegCeOece&: i. A B = B A [¬eÀceefvejHes#elee iegCeOece&] ii. A (B C) = (A B) C [meen®e³e& iegCeOece&] iii. A B A; A B B iv. A P; B P lej A B P

v. pej, A B lej, A B = A

pej, B A lej, A B = B

vi. A = DeeefCe A A = A

c. efJeefYevve meb®e mecepee, A = {2, 4, 6, 8} DeeefCe B = {1, 3, 5, 7}

A B = pesJne oesve meb®eele kegÀþuesner meeceeF&keÀ IeìkeÀ vemeleele, lesJne

l³ee meb®eebvee efJeefYevve meb®e cnCeleele.

A B

.2 .4

.6 .8

.1 .3

.5 .7

A B

.1 .3

.5

.7

.12 .9

A B

8


Jesve Deeke=Àleer®³ee meene³³eeves A DeeefCe B ns oesve efJeefYevve meb®e oμe&efJeues Deensle.

A B = efJeefYevve meb®eebmeeþer: i. A B = ii. A B iii. B A d. meb®eeb®ee meb³eesie :

meb®e A DeeefCe meb®e B efoues Demelee A DeeefCe B ceOeerue meJe& IeìkeÀ SkeÀ$e keÀªve le³eej Peeuesu³ee meb®eeme `meb³eesie meb®e' Demes cnCeleele. A DeeefCe B ³ee oesve meb®ee®ee meb³eesie ‘A B’ Demee oμe&efJeuee peelees Je l³ee®es Jee®eve À meb³eesie B' Demes keÀjleele.

mecepee, A = {1, 2, 3, 4, 5} DeeefCe B = {3, 5, 7, 9}

A B = {1, 2, 3, 4, 5, 7, 9} efkebÀJee A B = {x|x A efkebÀJee x B}

Jesve Deeke=Àleerleerue íe³eebefkeÀle Yeeie A B oμe&efJelees. meb³eesie meb®eeb®es iegCeOece&: i. A B = B A [¬eÀceefvejHes#elee iegCeOece&] ii. A (B C) = (A B) C [meen®e³e& iegCeOece&] iii. A A B DeeefCe B A B

iv. pej A B, lej A B = B DeeefCe pej B A, lej A B = A

v. A = A vi. A A = A efJelejCe iegCeOece&: i. A (B C) = (A B) (A C) ii. A (B C) = (A B) (A C) e. meb®ee®ee HetjkeÀ meb®e: pej U ne efJeμJemeb®e Je A ne l³ee®ee Ghemeb®e Deens, lej U

ceO³es Demeuesu³ee Hejbleg A ceO³es vemeuesu³ee meJe& IeìkeÀeb®ee meceeJesμe A ceO³es neslees. A uee A ®ee HetjkeÀmeb®e Demes cnCeleele.

lees A efkebÀJee Ac ves oμe&efJeuee peelees.

mecepee, U = {x|x ner vewmeefie&keÀ mebK³ee Demetve, x 9}

DeeefCe A = {1, 3, 5, 7}

³eeoer He×leerves, U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = { 2, 4, 6, 8, 9} efkebÀJee A = {x|x U DeeefCe x A}

Jesve Deeke=Àleerle meb®e A ®ee HetjkeÀ meb®e Heg{erueÒeceeCes oμe&efJeleele: ìerHe i. A A = ii. A A = U HetjkeÀ meb®ee®es iegCeOece& i. (A) = A ii. = U iii. U = iv. pej A B, lej B A

v. A A = vi. A A = U pej A DeeefCe B ns keÀesCelesner oesve meb®e Demeleerue lej i. (A B) = A B DeeefCe ii. (A B) = A B DeY³eeme 1.4 1. mecepee, P = {x|x ns ‘CATARACT’ ³ee μeyoeleerue

De#ej Deens} DeeefCe Q = {y|y ns ‘TRAC’³ee μeyoeleerue De#ej Deens}, lej efme× keÀje P = Q.

GkeÀue: meb®e P DeeefCe Q ³eeoer He×leerves KeeueerueÒeceeCes efueefnues

peeleele: P = {C, A, T, R} Q = {T, R, A, C} meb®e P DeeefCe Q SkeÀceskeÀeb®es GHemeb®e Deensle. lemes®e, meb®e P DeeefCe meb®e Q ceOeerue IeìkeÀ meejKes®e

Deensle. P = Q 2. Keeueerue Òel³eskeÀ GoenjCeeleerue meb®eeb®es meb³eesie meb®e

efuene. i. A = {2, 3, 5, 6, 7}, B = {4, 5, 7, 8} ii. C = {a, e, i, o, u}, D = {a, b, c, d}

A B

.1 A

.2

.4

.3 .7

.9

B

.5

.1 .3

.5 .7

.2

.6

.4

.8

.9

AA

U

9


iii. E = {x|x N DeeefCe x ne 12 ®ee efJeYeepekeÀ Deens.}

F = {y|y N DeeefCe y ne 18 ®ee efJeYeepekeÀ Deens}

GkeÀue: i. A = {2, 3, 5, 6, 7}, B = {4, 5, 7, 8} A B = {2, 3, 4, 5, 6, 7, 8} ii. C = {a, e, i, o, u}, D = {a, b, c, d} C D = {a, b, c, d, e, i, o, u} iii. meb®e E DeeefCe F ³eeoer He×leerves KeeueerueÒeceeCes efueefnues

peeleele: E = {1, 2, 3, 4, 6, 12} F = {1, 2, 3, 6, 9, 18} E F = {1, 2, 3, 4, 6, 9, 12, 18} 3. Keeueerue Òel³eskeÀ GoenjCeeleerue meb®eeb®es íso-meb®e

efuene. i. A = {1, 2, 4, 5, 7}, B = {2, 3, 4, 8} ii. C = {x|x N, 5 < x 10}, D = {y|y W, 5 y < 10} iii. E = {x|x I, x < 0}, F = {y|y I, y > 0} GkeÀue: i. A = {1, 2, 4, 5, 7}, B = {2, 3, 4, 8} A B = {2, 4} ii. meb®e C DeeefCe D ³eeoer He×leerves KeeueerueÒeceeCes efueefnues

peeleele: C = {6, 7, 8, 9, 10} D = {5, 6, 7, 8, 9} C D = {6, 7, 8, 9} iii. meb®e E DeeefCe F ³eeoer He×leerves KeeueerueÒeceeCes efueefnues

peeleele: E = {… , 4, 3, 2, 1} F = {1, 2, 3, 4, …} E F = { } efkebÀJee 4. mecepee, U = {x|x = 2n, n W, n < 8} ne efJeμJemeb®e

Deens. A = {y|y = 4n, n W, n < 4} B = {z|z = 8n,

n W, n 2} lej Keeueerue meb®e efuene. i. A ii. B iii. (A B) iv. (A B)

GkeÀue: meb®e U, A DeeefCe B ³eeoer He×leerves KeeueerueÒeceeCes efueefnues

peeleele: U = {20, 21, 22, 23, 24, 25, 26, 27} = {1, 2, 4, 8, 16, 32, 64, 128} A = {40, 41, 42, 43} = {1, 4, 16, 64} B = {80, 81, 82} = {1, 8, 64} i. A = {2, 8, 32, 128} ii. B = {2, 4, 16, 32, 128} iii. A B = {1, 4, 8, 16, 64} (A B) = {2, 32, 128} iv. (A B) = {1, 64} (A B) = {2, 4, 8, 16, 32, 128} 5. mecepee A = {a|a ns ‘college’³ee μeyoeleerue De#ej

Deens} DeeefCe B = {b|b ns ‘luggage’³ee μeyoeleerue De#ej Deens }Je U = {a, b, c, d, e, f, g, l, o, u}.

lej oeKeJee keÀer, i. (A B) = A B ii. (A B) = A B efme×lee: i. ³eeoer He×leerÒeceeCes meb®e A DeeefCe B KeeueerueÒekeÀejs efueefnues

peeleele: A = {c, o, l, e, g} B = {l, u, g, a, e} U = {a, b, c, d, e, f, g, l, o, u} A = {a, b, d, f, u} B = {b, c, d, f, o} A B = {a, c, e, g, l, u, o} [eJeer yeepet = (A B) = {b, d, f} .… (i)

GpeJeer yeepet = A B = {b, d, f} .… (ii)

(i) DeeefCe (ii) Jeªve, [eJeer yeepet = GpeJeer yeepet (A B) = A B ii. A B = {l, g, e} [eJeer yeepet = (A B) = {a, b, c, d, f, o, u}

.… (iii) GpeJeer yeepet = A B = {a, b, c, d, f, o, u}

…. (iv) (iii) DeeefCe (iv) Jeªve, [eJeer yeepet = GpeJeer yeepet (A B) = A B

10

F. 9 Jeer : yeerpeieefCele 1.8 meb®eeleerue IeìkeÀeb®eer mebK³ee pej meb®e A ne keÀesCeleener SkeÀ meb®e Demesue lej l³eeleerue IeìkeÀeb®eer mebK³ee n (A) Deμeer oμe&efJeleele. Goe: i. mecepee, A = {x|x N, 7 < x 12} A = {8, 9, 10, 11, 12} n(A) = 5 ii. efjkeÌle meb®eemeeþer, n() = 0 iii. n(A B) = n(A) + n(B) n(A B) ³ee efvel³emeceeveles®ee He[leeUe IesC³eemeeþer, Keeueerue GoenjCe ie=nerle Oeªve: A = {2, 3, 4} DeeefCe B = {3, 4, 5, 6}

A B = {2, 3, 4, 5, 6} DeeefCe A B = {3, 4} n(A) = 3, n(B) = 4, n(A B) = 5 DeeefCe

n(A B) = 2 [eJeer yeepet = n(A B) = 5 .... (i)

GpeJeer yeepet = n(A) + n(B) n(A B)

= 3 + 4 2 = 5 .... (ii) n(A B) = n(A) + n(B) n(A B) .... [(i) DeeefCe (ii) Jeªve ] DeY³eeme 1.5 1. pej A = {1, 3, 5, 6, 7}, B = {4, 6, 7, 9}, lej

Keeueerue efvel³emeceevelee He[leeUtve Hene. n (A B) = n(A) + n(B) n(A B) efme×lee: A = {1, 3, 5, 6, 7} DeeefCe B = {4, 6, 7, 9}

A B = {1, 3, 4, 5, 6, 7, 9} DeeefCe A B = {6, 7} n(A) = 5, n(B) = 4, n(A B) = 7 DeeefCe

n(A B) = 2 [eJeer yeepet = n(A B) = 7 ....(i)


= 5 + 4 2 = 7 ....(ii) [eJeer yeepet = GpeJeer yeepet ....[(i) DeeefCe (ii) Jeªve] n(A B) = n(A) + n(B) n (A B)

2. pej n(A) = 5, n(A B) = 9, n(A B) = 2 lej n(B) = efkeÀleer?

GkeÀue: efouesues, n(A) = 5, n(A B) = 9, n(A B) = 2

n(B) = ? efvel³emeceevelesvegmeej,

n(A B) = n(A) + n(B) n(A B) 9 = 5 + n(B) 2 9 – 5 + 2 = n(B) n(B) = 6 3. SkeÀe μeeues³e Jemeefleie=nele 100 efJeÐeeLeea Deensle.

l³eeHewkeÀer 60 efJeÐeeLeeaa ®ene efHeleele, 50 efJeÐeeLeeaa keÀe@HeÀer efHeleele DeeefCe 30 efJeÐeeLeeaa ®ene Je keÀe@HeÀer ns oesvner ÒekeÀej®es Hes³e efHeleele. lej SkeÀner Hes³e ve efHeCeeN³ee efJeÐeeL³ee¥®eer mebK³ee keÀe{e.

GkeÀue: mecepee, U ne Jemeefleie=neleerue efJeÐeeL³ee¥®ee efJeμJemeb®e Deens, meb®e T ne ®ene efHeCeeN³ee efJeÐeeL³ee¥®ee meb®e Deens DeeefCe C

ne keÀe@HeÀer efHeCeeN³ee efJeÐeeL³ee¥®ee meb®e Deens. n(U) = 100, n(T) = 60, n(C) = 50, n(T C) = 30 efvel³emeceevelesvegmeej, n(T C) = n(T) + n(C) n(T C) = 60 + 50 30 = 110 30 n(T C) = 80 80 efJeÐeeLeeaa ns keÀe@HeÀer efkebÀJee ®ene efkebÀJee oesvner Hes³e efHeleele,

Hejbleg Jemeefleie=nele 100 efJeÐeeLeeaa Deensle. ®ene efkebÀJee keÀe@HeÀer ns SkeÀner Hes³e ve efHeCeeN³ee efJeÐeeL³ee¥®eer

mebK³ee = n(U) n(T C) = 100 80 = 20 SkeÀner Hes³e ve efHeCeeN³ee efJeÐeeL³ee¥®eer mebK³ee 20 Deens. 4. 110 efJeÐeeL³ee¥vee efveUe efkebÀJee iegueeyeer ³eeHewkeÀer

DeeJe[lee jbie keÀesCelee ns efJe®eejC³eele Deeues. Òel³eskeÀ efJeÐeeL³ee&®ee SkeÀ lejer jbie DeeJe[lee neslee. l³eeHewkeÀer 60 efJeÐeeL³ee¥veer efveUe jbie lej 70 efJeÐeeL³ee¥veer iegueeyeer jbie DeeJe[le Demeu³ee®es meebefieleues. lej efkeÀleer efJeÐeeL³ee¥veer oesvner jbie DeeJe[le Demeu³ee®es meebefieleues?

GkeÀue: mecepee, efveUe jbie DeeJe[CeeN³ee efJeÐeeL³ee¥®eer mebK³ee n(B) DeeefCe

iegueeyeer jbie DeeJe[CeeN³ee efJeÐeeL³ee¥®eer mebK³ee n(P) Deens.

11


n(B) = 60 DeeefCe n(P) = 70

iegueeyeer efkebÀJee efveUe jbie DeeJe[Ceejs SketÀCe efJeÐeeLeea n(B P) = 110 efvel³emeceevelesvegmeej, n(B P) = n(B) + n(P) n(B P)

110 = 60 + 70 n(B P)

n(B P) = 60 + 70 110

n(B P) = 20

oesvner jbie DeeJe[le Demeuesu³ee efJeÐeeL³ee¥®eer mebK³ee 20

Deens. 5. efouesu³ee Deeke=Àleer®es efvejer#eCe keÀje DeeefCe Keeueerue

meceevelee He[leeUtve Hene. n(A B C) = n(A) + n(B) + n(C) n(A B)

n(B C) n(C A) + n(A B C) efme×lee: [eJeer yeepet = n(A B C)

A B C = {1, 2, 3, 4, 5, 6, 7, 8, 9} n(A B C) = 9 …. (i) Deelee, A = {1, 2, 3, 4, 5} n(A) = 5

B = {2, 3, 6, 7, 8} n(B) = 5

C = {3, 4, 6, 9} n(C) = 4

A B = {2, 3} n(A B) = 2

B C = {3, 6} n(B C) = 2

C A = {3, 4} n(C A) = 2

A B C = {3} n(A B C) = 1

GpeJeer yeepet = n(A) + n(B) + n(C) n(A B)

n(B C) n(C A) + n(A B C)

= 5 + 5 + 4 2 2 2 + 1 = 9 …. (ii) [eJeer yeepet = GpeJeer yeepet ....[(i) Je (ii) Jeªve] n(A B C) = n(A) + n(B) + n(C)

n(A B) n(B C) n(C A)

+ n(A B C)

Òeμvemeb®e - 1 1. KeeueerueHewkeÀer keÀesCeles mecetn ns `meb®e' Deensle? i. legce®³ee efpeunîeeleerue Þeerceble J³ekeÌleeR®ee mecetn. ii. 50 Hes#ee ueneve Demeuesu³ee vewmeefie&keÀ mebK³eeb®ee

mecetn. iii. Yeejleeleerue meJee&le yegef×ceeve J³ekeÌleeR®ee mecetn. iv. Heefnu³ee one cetU mebK³eeb®ee mecetn. v. ‘T’ De#ejeves megª nesCeeN³ee Fbûepeerleerue

Jeejeb®ee mecetn. vi. Je<ee&leerue keÀener ceefnv³eeb®ee mecetn. vii. legce®³ee μeeues³e ûebLeeue³eeleerue meJe& HegmlekeÀeb®ee

mecetn. viii. legce®³ee Jeiee&leerue ®egCe®egCeerle efJeÐeeL³ee¥®ee

mecetn. ix. meele®³ee Heìerleerue mebK³eeb®ee mecetn. x. ÒeLece IeìkeÀ ®ee®eCeerle KetHe iegCe efceUefJeuesu³ee

legce®³ee Jeiee&leerue efJeÐeeL³ee¥®ee mecetn.

GkeÀue: efouesu³ee mecetnebHewkeÀer (ii), (iv), (v), (vii) DeeefCe (ix) ns meb®e Deensle. GJe&efjle mecetn ns meb®e veenerle keÀejCe ³ee yeeyeer meeHes#e Deensle Je l³ee J³eeqkeÌleHejlJes yeoueleele. 2. Keeueerue meb®e ³eeoer He×leerves efuene. i. A = {x|x I, x W} ii. B = {x|x ner oesve DebkeÀer mebK³ee Demetve efle®³ee

DebkeÀeb®ee iegCeekeÀej 10 ®³ee Heìerle Deenss.}

iii. C = {x|x ner mebK³ee 120 ®eer cetU DeJe³eJe Deens }

iv. D = {x|x I DeeefCe x2 < 10}

v. E = 2

n,2 n 4,n N

n 1

x x

GkeÀue: i. A = {… , 3, 2, 1} ii. B = {25, 45, 52, 54, 56, 58, 65, 85} iii. C = {2, 3, 5} iv. D = {3, 2, 1, 0, 1, 2, 3} v. 2 n 4

n = 2, 3, 4

n = 2 meeþer, 2

n

n 1 =

2

2

(2) 1=

2

3

.1

.5

.2

.3 .4

.9

.6

.7

.8 B A

C

12


n = 3 meeþer, 2

n

n 1=

2

3

(3) 1=

3

8

n = 4 meeþer, 2

n

n 1 =

2

4

(4) 1 =

4

15

E = 2 3 4

, ,3 8 15

3. Keeueerue meb®e iegCeOece& He×leerves efuene. i. F = {I, N, D, A} ii. G = {1, 1} iii. H = {3, 9, 27, 81, 243} iv. J = {15, 24, 33, 42, 51, 60}

v. K =

1 2 3 4 5, , , ,

2 5 10 17 26

GkeÀue: i. F = {x|x ns ‘INDIA’ ³ee μeyoeleerue De#ej Deens}

ii. G = {y| y ®ee Jeie& 1 Deens.}

efkebÀJee G = {y|y ns 1 ®es Jeie&cetU Deens.}

iii. H = {a|a = 3n, n N, n 5} iv. J = {b|b ner DebkeÀeb®eer yesjerpe 6 DemeCeejer oesve DebkeÀer

mebK³ee Deens}

v. pesJne n = 1, c = 2

1 1

2(1) 1

pesJne n = 2, c = 2

2 2

5(2) 1

pesJne n = 3, c = 2

3 3

10(3) 1

pesJne n = 4, c = 2

4 4

17(4) 1

pesJne n = 5, c = 2

5 5

26(5) 1

K = 2

nc c = , n N, n 5

n +1

4. Keeueerue meb®eeb®es SkeÀ IeìkeÀ meb®e efkebÀJee èfjkeÌle meb®e'

Demes JeieeakeÀjCe keÀje. i. A = {x|x ner $eÝCe vewmeefie&keÀ mebK³ee Deens}

ii. B = {y|y ner 4 Hes#ee ueneve DemeCeejer efJe<ece cetU mebK³ee Deens} iii. C = {z|z ner vewmeefie&keÀ mebK³ee Demetve, 5 < z < 7} iv. D = {d|d N, d2 0}

GkeÀue: i. Òel³eskeÀ vewmeefie&keÀ mebK³ee ner Oeve Demeles. A = { } ne efjkeÌle meb®e Deens. ii. B = {3} ne SkeÀ IeìkeÀ meb®e Deens.

iii C = {6} ne SkeÀ IeìkeÀ meb®e Deens.

iv. vewmeefie&keÀ mebK³es®ee Jeie& ne μetv³eeHes#ee keÀceer efkebÀJee μetv³e Demet μekeÀle veener.

D = { } ne efjkeÌle meb®e Deens. 5. Keeueerue mebK³eeb®es `meeble' efkebÀJee `Deveble' Demes

Jeiee&rkeÀjCe keÀje. i. A = {x|x ner 3 ®³ee Heìerleerue mebK³ee Deens}

ii. B = {y|y ne 13 ®ee DeJe³eJe Deens}

iii. C = {…, 3, 2, 1, 0} iv. D = {x|x = 2n, n N} GkeÀue: i. A = {3, 6, 9, 12, …} ne Deveble meb®e Deens. ii. B = {1, 13} ne meeble meb®e Deens. iii. C ne Deveble meb®e Deens. iv. D = {20, 21, 22, 23, 24,...} = {2, 4, 8, 16, 32, …} D ne Deveble meb®e Deens. 6. KeeueerueHewkeÀer keÀesCeles meb®e meceeve Deensle? i. N = {1, 2, 3, 4, …} ii. W = {0, 1, 2, 3, …} iii. A = {x|x = 2n, n W} iv. B = W {0} GkeÀue: i. N ={1, 2, 3, 4, …} ii. W = {0, 1, 2, 3, …} iii. A = {20, 21, 22, 23, …} = {1, 2, 4, 8, …} iv. B = W {0} = {1, 2, 3, 4, …} ³esLes, meb®e N DeeefCe B SkeÀceskeÀeb®es GHemeb®e Deensle. meb®e N DeeefCe meb®e B ceOeerue IeìkeÀ meceeve Deensle. N = B

13


7. pej A = {7, 5, 2} DeeefCe B = 3 125, 4, 49 .

meb®e A DeeefCe B ns meceeve Deensle keÀe³e? legce®³ee GÊeje®es mHe<ìerkeÀjCe keÀje.

GkeÀue: A = {7, 5, 2},

B = 3 125, 4, 49

B = {5, 2, 2, 7, 7} ³esLes, A ne B ®ee GHemeb®e Deens Hejbleg B ne A ®ee GHemeb®e

veener. meb®e A DeeefCe meb®e B ceOeerue IeìkeÀ meceeve veenerle.

A B 8. pej A = {1, 2, 3, 4}, B = {2, 4, 6, 8},

C = {3, 4, 5, 6} DeeefCe U = {x|x N, x < 10} ne efJeμJemeb®e Deens. lej Keeueerue iegCeOece& He[leeUtve Hene.

i. A (B C) = (A B) C ii. A (B C) = (A B) (A C) iii. A (C) = (A B) (A C) iv. (A B) = A B v. (A B) = A B vi. (A) = A GkeÀue: meb®e U ³eeoer He×leerves KeeueerueÒeceeCes efueefnlee ³eslees. U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 3, 4} B = {2, 4, 6, 8} C = {3, 4, 5, 6} i. B C = {2, 3, 4, 5, 6, 8} A B = {1, 2, 3, 4, 6, 8} [eJeer yeepet = A (B C)

= {1, 2, 3, 4, 5, 6, 8} …. (i) GpeJeer yeepet = (A B) C

= {1, 2, 3, 4, 5, 6, 8} …. (ii) (i) Je (ii) Jeªve, [eJeer yeepet = GpeJeer yeepet

A (B C) = (A B) C ii. B C = {4, 6} A B = {1, 2, 3, 4, 6, 8} A C = {1, 2, 3, 4, 5, 6} [eJeer yeepet = A (B C)

= {1, 2, 3, 4, 6} …. (i) GpeJeer yeepet = (A B) (A C)

= {1, 2, 3, 4, 6} …. (ii)

(i) Je (ii) Jeªve, [eJeer yeepet = GpeJeer yeepet A (B C) = (A B) (A C) iii. B C = {2, 3, 4, 5, 6, 8} [eJeer yeepet = A (B C)

= {2, 3, 4} …. (i) A B = {2, 4} A C = {3, 4} GpeJeer yeepet = (A B) (A C)

= {2, 3, 4} .… (ii) (i) DeeefCe (ii) Jeªve, [eJeer yeepet = GpeJeer yeepet

A (C) = (A B) (A C) iv. A B = {1, 2, 3, 4, 6, 8} (A B) = {5, 7, 9} A = {1, 2, 3, 4} A = {5, 6, 7, 8, 9} B = {2, 4, 6, 8} B = {1, 3, 5, 7, 9} [eJeer yeepet = (A B) = {5, 7, 9} …. (i) GpeJeer yeepet = A B = {5, 7, 9} …. (ii) (i) DeeefCe (ii) Jeªve, [eJeer yeepet = GpeJeer yeepet

(A B) = A B v. A B = {2, 4} (A B) = {1, 3, 5, 6, 7, 8, 9} A = {5, 6, 7, 8, 9} B = {1, 3, 5, 7, 9} [eJeer yeepet = (A B) = {1, 3, 5, 6, 7, 8, 9} …. (i) GpeJeer yeepet = A B = {1, 3, 5, 6, 7, 8, 9} …. (ii) (i) DeeefCe (ii) Jeªve, [eJeer yeepet = GpeJeer yeepet

(A B) = A B vi. A = {5, 6, 7, 8, 9} [eJeer yeepet = (A) = {1, 2, 3, 4} …. (i) GpeJeer yeepet = A

= {1, 2, 3, 4} …. (ii) (i) DeeefCe (ii) Jeªve, [eJeer yeepet = GpeJeer yeepet

(A) = A

14


9. Keeueerue meb®e ns èfjkeÌle meb®e' Deensle efkebÀJee veenerle les mekeÀejCe meebiee.

i. A = {x|x I, x2 ner Oeve mebK³ee veener}

ii. B = {b|b N, 2b + 1 ner mece mebK³ee Deens}

iii. C = {c|c N, c ner efJe<ece mebK³ee Demetve c2 mece Deens} GkeÀue: i. keÀesCel³eener Oeve efkebÀJee $eÝCe mebK³es®ee Jeie& ne vesnceer Oeve

Demelees. keÀesCel³eener HetCee¥keÀ mebK³es®ee Jeie& $eÝCe Demet μekeÀle veener.

³eeuee DeHeJeeo μetv³e ne Deens; p³ee®ee Jeie& Oeve efkebÀJee $eÝCe vemelees.

A = {0} meb®e A ne efjkeÌle meb®e veener. ii. ³esLes, 2b ner mece mebK³ee Deens DeeefCe 1 ner efJe<ece mebK³ee

Deens. Hejbleg, mece mebK³ee DeeefCe efJe<ece mebK³ee ³eeb®eer yesjerpe vesnceer

efJe<ece mebK³ee ³esles. 2b + 1, ®eer efkebÀcele, b N ®³ee keÀesCel³eener efkeÀceleermeeþer

efJe<ece mebK³ee Demeles. meb®e B ne efjkeÌle meb®e Deens. iii. efJe<ece mebK³es®ee Jeie& vesnceer efJe<ece mebK³ee®e Demeles. c2 keÀOeerner mece mebK³ee ³esCeej veener. meb®e C ne efjkeÌle meb®e Deens. 10. iegCeOece& He×leerves efueefnlee ³esleerue Hejbleg ³eeoer He×leerves

efueefnlee ³esCeej veener Deμee SkeÀe meb®ee®es GoenjCe Ðee.

GkeÀue: mecepee ‘Q’ ne Heefjces³e mebK³eeb®ee meb®e Deens. lees iegCeOece& He×leerves KeeueerueÒeceeCes efueefnlee ³eslees:

Q = a

a I, b I b 0b

DeeeCf e

Hejbleg, ne meb®e ³eeoer He×leerves efueefnlee ³esCeej veener. 11. Keeueerue meb®eeb®es μekeÌ³e nesleerue lesJe{s meJe& GHemeb®e

efuene: i. ii. A = {1} iii. B = {1, 2} iv. C = {a, b, c, d}

GkeÀue: i. efjkeÌle meb®ee®ee SkeÀ®e GHemeb®e Demelees, lees cnCepes

ii. meb®e A ®es GHemeb®e efjkeÌle meb®e { } DeeefCe meb®e A mJele: Deensle. cnCepes®e DeeefCe {1}

iii. meb®e B ®es meJe& μekeÌ³e GHemeb®e: , {1}, {2}, {1, 2} iv. C meb®ee®es meJe& μekeÌ³e GHemeb®e: {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d},

{b, c}, {b, d}, {c, d}, {a, b, c}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c, d}

12. Keeueerue meb®eeb®es μekeÌ³e nesleerue lesJe{s meJe& ³egkeÌle

(proper) GHemeb®e efuene. i. A = {a, b} ii. B = {a, b, c} GkeÀue: i. A ®es ³egkeÌle GHemeb®e {a} DeeefCe {b} Deensle. ii. B ®es ³egkeÌle GHemeb®e {a}, {b}, {c}, {a, b}, {b, c},

{c, a} ns Deensle. 13. meb®e A DeeefCe B Demes le³eej keÀje keÀer, A Je B ns meeble

meb®e Demetve les HejmHejeb®es efJeYekeÌle meb®e (disjoint) Deensle.

GkeÀue: mecepee, B = {1, 3, 5, 7}, A = {2, 4, 6, 8} ns meeble

meb®e Deensle Hejbleg l³eeb®ee ísomeb®e efjkeÌle meb®e Deens. cnCepes®e A B =

A DeeefCe B ns efJeYekeÌle meb®e Deensle. 14. mecepee A = {a, b, c, d}, B = {a, b, c},

C = {b, d, e} lej Keeueerue Deìer HeeUCeejs meb®e D Je meb®e E μeesOee.

i. D A, D B ii. C E, B E = GkeÀue: i. ³esLes, D A DeeefCe D B

D = {a, b}/{a, c}/{a, d}/{a, b, d} /{b, c, d}/{a, c, d}

ii. ³esLes, C E DeeefCe B E =

meb®e E ceO³es meb®e B ®ee SkeÀner IeìkeÀ vemeuee Heeefnpes. meb®e C ne meb®e E ®ee Deeflemeb®e Deens.

E = {d}/{e}/{d, e}

15


15. pej U = {x|x N, x < 10};

A = {a|a U ner mece mebK³ee Deens},

B = {b|b U ner 6 ®eer efJeYeepekeÀ Deens}. lej n(A) + n(B) = n(A B) + n(A B) ³ee met$ee®ee

He[leeUe I³ee. GkeÀue: meb®e U DeeefCe meb®e A ³eeoer He×leerves KeeueerueÒeceeCes efueefnues

peeleele: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {2, 4, 6, 8} ….[efJeμJemeb®eeleerue mece mebK³ee] n(A) = 4 meb®e B ³eeoer He×leerves KeeueerueÒeceeCes efueefnuee peelees. B = {1, 2, 3, 6} ….[ b U]

n(B) = 4 A B = {1, 2, 3, 4, 6, 8} n(A B) = 6 A B = {2, 6} n(A B) = 2 [eJeer yeepet = n(A) + n(B)

= 4 + 4 = 8 ....(i) GpeJeer yeepet = n(A B) + n(A B)

= 6 + 2 = 8 ....(ii) (i) Je (ii), Jeªve, [eJeer yeepet = GpeJeer yeepet n(A) + n(B) = n(A B) + n(A B) 16. SkeÀe Jeiee&leerue efJeÐeeL³ee¥HewkeÀer 50 efJeÐeeLeeaa Fbûepeerle

GÊeerCe& Deensle, 60 efJeÐeeLeeaa ieefCeleele GÊeerCe& Deensle, oesvner efJe<e³eele 40 efJeÐeeLeeaa GÊeerCe& Deensle lej oesvnerHewkeÀer efkeÀceeve SkeÀe efJe<e³eele efkeÀleer efJeÐeeLeea GÊeerCe& Peeues Deensle?

GkeÀue: mecepee, FbeqiueμeceO³es Heeme Peeuesu³ee efJeÐeeL³ee¥®eer mebK³ee E

meb®e oμe&efJelees. n(E) = 50 ieefCeleele Heeme Peeuesu³ee efJeÐeeL³ee¥®eer mebK³ee M meb®e

oμe&efJelees. n(M) = 60 ieefCele Je FbeqiueμeceO³es Heeme Peeuesu³ee efJeÐeeL³ee¥®eer mebK³ee = n(M E) = 40

efvel³emeceevelesJeªve, n(E M) = n(E) + n(M) n(E M) = 50 + 60 – 40 = 110 40 = 70 oesvnerHewkeÀer efkeÀceeve SkeÀe efJe<e³eele (Fbeqiueμe efkebÀJee

ieefCele) 70 efJeÐeeLeea GÊeerCe& Peeues Deensle. 17. otjoμe&ve®³ee SkeÀe meJex#eCeevegmeej 136 efJeÐeeLeeaa HeÀkeÌle

P1 ne keÀe³e&¬eÀce Heenleele, 107 efJeÐeeLeeaa HeÀkeÌle P2 ne keÀe³e&¬eÀce Heenleele. 27 efJeÐeeLeeaa HeÀkeÌle P3 ne keÀe³e&¬eÀce Heenleele, 25 efJeÐeeLeeaa HeÀkeÌle P1 Je P2 ne keÀe³e&¬eÀce Heenleele. Hejbleg P3 keÀe³e&¬eÀce Heenele veenerle.

37 efJeÐeeLeeaa P2 Je P3 ns keÀe³e&¬eÀce Heenleele Hejbleg P1

ne keÀe³e&¬eÀce Heenele veenerle. 53 efJeÐeeLeeaa P1 Je P3 ns keÀe³e&¬eÀce Heenleele Hejbleg P2 ne keÀe³e&¬eÀce Heenele veenerle. 40 efJeÐeeLeeaa ns P1, P2 Je P3 ns eflevner keÀe³e&¬eÀce Heeneleele. pej 80 efJeÐeeLeeaa SkeÀner keÀe³e&¬eÀce Heenele vemeleerue lej Jesve-Deeke=Àleer®³ee meene³³eeves Keeueerue Òeμve mees[Jee:

i. P1 keÀe³e&¬eÀce HeenCeeN³eeb®eer mebK³ee keÀe{e.

ii. P2 efkebÀJee P3 keÀe³e&¬eÀce HeenCeeN³eeb®eer mebK³ee keÀe{e.

iii. efkeÀleer efJeÐeeL³ee¥®ee meJex#eCeele meceeJesμe neslees? GkeÀue: efouesueer ceeefnleer Jesve Deeke=Àleer®³ee meene³³eeves KeeueerueÒeceeCes

oμe&efJelee ³esles. i. Jesve Deeke=Àleervegmeej,

P1 ne keÀe³e&¬eÀce HeenCeeN³eeb®eer mebK³ee = n(P1)

= 136 + 25 + 53 + 40 = 254 P1 ne keÀe³e&¬eÀce HeenCeejs 254 peCe Deensle.

P2 P1

P3

27

136

53

25 107

37 40

80

U

16


ii. P2 efkebÀJee P3 keÀe³e&¬eÀce HeenCeeN³eeb®eer mebK³ee = n(P2 P3) = 107 + 25 + 40 + 37 + 53 + 27 = 289 P2 efkebÀJee P3 keÀe³e&¬eÀce HeenCeejs 289 peCe Deensle. iii. meJex#eCeele meceeefJe<ì Demeuesues SketÀCe efJeÐeeLeea = HeÀkeÌle P1 ne keÀe³e&¬eÀce HeenCeeN³eeb®eer mebK³ee + P2 efkebÀJee

P3 keÀe³e&¬eÀce HeenCeeN³eeb®eer mebK³ee + 80

= 136 + 289 + 80 = 505 meJex#eCeele meceeefJe<ì Demeuesues SketÀCe efJeÐeeLeea 505. 18. Keeueerue JeCe&ve DemeCeejs meb®e A Je B Demet μekeÀCeej

veenerle ns efme× keÀje. n(A) = 32, n(B) = 42, n (A B) = 12, n (A B) = 64 GkeÀue: efouesues n(A) = 32, n(B) = 42, n(A B) = 12

DeeefCe n(A B) = 64

efvel³emeceevelesJeªve, n(A B) = n(A) + n(B) n(A B), [eJeer yeepet = n(A B) = 64 ....(i)


= 32 + 42 12 = 62 ....(ii) (i) Je (ii), Jeªve, [eJeer yeepet GpeJeer yeepet efouesu³ee JeCe&vee®es meb®e A DeeefCe B Demet μekeÀCeej

veenerle. 19. A ne meb®e legce®³ee μeeUsleerue cegueeb®ee meb®e Demetve meb®e

B ne legce®³ee μeeUsleerue cegueeR®ee meb®e Deens. pej U ne legce®³ee μeeUsleerue meJe& efJeÐeeL³ee¥®ee meb®e Demesue DeeefCe C ne legce®³ee μeeUsleerue KesUele Yeeie IesCeeN³ee efJeÐeeL³ee¥®ee meb®e Demesue lej Keeueerue meb®eeb®es μeyoele JeCe&ve keÀje DeeefCe les meb®e Jesve-Deeke=Àleerves oeKeJee:

i. B C ii. A (B C) GkeÀue: i. B C cnCepes KesUele menYeeieer nesCeeN³ee cegueeR®ee meb®e

ii. A (B C) ne meb®e meJe& cegues efkebÀJee KesUele menYeeieer nesCeeN³ee cegueer oμe&Jelees.

20. A, B DeeefCe C Demes meb®e Deensle keÀer, A B,

A C = DeeefCe B C lej Jejerue meb®e A, B DeeefCe C ns Jesve-Deeke=Àleer®³ee meene³³eeves oeKeJee DeeefCe A (B C) ne meb®e íe³eebefkeÀle Yeeieeves oeKeJee.

GkeÀue: i. A B meb®e A ne meb®e B ®ee ³egkeÌle GHemeb®e Deens. cnCepes®e, meb®e A ne meb®e B ®³ee Deeleerue Yeeieele Deens. ii. A C = meb®e A DeeefCe meb®e C ísole veenerle. iii. B C meb®e B DeeefCe meb®e C ísoleele. 21. A, B Je C Demes meb®e Deensle keÀer A B ,

B C DeeefCe A C , lej A B C Demee efve<keÀ<e& keÀe{lee ³esF&ue

keÀe³e? legce®³ee GÊeje®es mHe<ìerkeÀjCe keÀje.

GkeÀue: ³esLes oesve μekeÌ³elee Deensle: i. pej A = {a, b}

B = {b, c} C = {c, a} lej Demes efometve ³esles keÀer A B ,

B C , C A , Hejbleg, A B C =

C B A

B C

A C B

A (B C)

B C

A

B C

A (B C)

B C

17


ii. pej A = {a, b}

B = {a, c} C = {a, b, c} lej Demes efometve ³esles keÀer A B ,

B C , C A , Hejbleg, A B C = {a}

DeeHeCe A B C Demee efve<keÀ<e& keÀe{t μekeÀle veener.

22. meb®e A, B Je C ³eeb®³ee ³eesi³e l³ee GoenjCeeb®³ee

meene³³eeves Keeueerue efJeOeeve He[leeUtve Heene. ‘pej A B, B C, lej A C’

GkeÀue: mecepee,A = {x, y, z}, B = {a, x, y}, C = {y, w}

A meb®eeleerue Òel³eskeÀ IeìkeÀ meb®e B ceO³es veener. A B meb®e B ®ee Òel³eskeÀ IeìkeÀ meb®e C ceO³es veener. B C meb®e A ®ee Òel³eskeÀ IeìkeÀ meb®e C ceO³es veener. A C 23. pej A DeeefCe B ns keÀesCelesner oesve meb®e Demeleerue, lej

efme× keÀje keÀer, i. (A B) = A B ii. (A B) = A B [ìerHe: (A B) A B oeKeJee DeeefCe

³eeGueìmeg×e] GkeÀue:

mecepee, U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 3, 4, 5} B = {3, 4, 5, 6, 7} i. A B = {1, 2, 3, 4, 5, 6, 7} [eJeer yeepet = (A B) = {8} …. (i)

A = {6, 7, 8} B = {1, 2, 8} GpeJeer yeepet = A B = {8} …. (ii)

(i) DeeefCe (ii) Jeªve, [eJeer yeepet = GpeJeer yeepet (A B) = A B ii. Deelee, A B = {3, 4, 5} [eJeer yeepet = (A B) = {1, 2, 6, 7, 8} …. (i)

GpeJeer yeepet = A B = {1, 2, 6, 7, 8} …. (ii)

(i) DeeefCe (ii) Jeªve, [eJeer yeepet = GpeJeer yeepet (A B) = A B SkeÀe iegCee®es Òeμve 1. Keeueerue meb®e iegCeOece& He×leerves efuene. A = {2, 3, 5, 7, 11, 13, 17} GkeÀue: meb®e A iegCeOece& He×leerves KeeueerueÒeceeCes efueefnlee ³eslees: A = {x| x ner cetU mebK³ee Deens, x < 18} 2. Keeueerue meb®e ³eeoer He×leerves efuene.

B = {x/x ner vewmeefie&keÀ mebK³ee Deens DeeefCe 4 x < 10

GkeÀue: meb®e B ³eeoer He×leerves KeeueerueÒeceeCes efueefnlee ³eslees: B = {4, 5, 6, 7, 8, 9} 3. pej A = {1, 2, 3, 4, 5, 6} DeeefCe B = {1, 3, 5, 7}

lej A B meeþer Jesve ef®e$e keÀe{e. GkeÀue: 4. pej U = {1, 2, 3, 4, 5, 6, 7, 8, 9} ne efJeμJemeb®e

Demesue DeeefCe C = {5, 6, 7, 8} lej C keÀe{e. GkeÀue: efouesues, U = {1, 2, 3, 4, 5, 6, 7, 8, 9} DeeefCe C = {5, 6, 7, 8}

C = {1, 2, 3, 4, 9} 5. pej A = {9, 11, 13, 15} DeeefCe B = {1, 3, 5, 7}

lej A B keÀe{e. GkeÀue: efouesues, A = {9, 11, 13, 15} DeeefCe B = {1, 3, 5, 7}

A B = { } efkebÀJee

A B

7 1

53

246

A B

U

18


6. pej A DeeefCe B Demes meb®e Deensle keÀer, n(B) = 8,

n(A B) = 11, n(A B) = 6, lej n(A) keÀe{e. GkeÀue: efvel³emeceevelesvegmeej,

n (A B) = n(A) + n(B) n(A B) 11 = n(A) + 8 6 n(A) = 11 2 n(A) = 9 7. KeeueerueHewkeÀer keÀesCeles meb®e meceeve Deensle les meebiee.

A = {x|x W, x < 6} B = {1, 2, 3, 4, 5, 6} C = {0, 1, 2, 3, 4, 5}

GkeÀue: ³esLes, A = {0, 1, 2, 3, 4, 5} DeeefCe C = {0, 1, 2, 3, 4, 5}

A = C 8. Keeueerue meb®eeb®es `SkeÀ IeìkeÀ meb®e' efkebÀJee èfjkeÌle

meb®e' Demes JeieeakeÀjCe keÀje. i. A = {x~x ner vewmeefie&keÀ mebK³ee Deens, x 5

DeeefCe x 7}

ii. B = { x~x ner mece cetU mebK³ee Deens.}

GkeÀue: i. x < 5 DeeefCe x > 7 meeþer SkeÀner meeceeF&keÀ mebK³ee

veener. A = {}

meb®e A ne efjkeÌle meb®e Deens.

ii B = {2} meb®e B ne SkeÀ IeìkeÀ meb®e Deens. 9. pej A = {2, 3, 4, 5} DeeefCe B = {1, 2, 5, 6}, lej

A B μeesOee. GkeÀue: efouesues, A = {2, 3, 4, 5} DeeefCe B = {1, 2, 5, 6} A B = {1, 2, 3, 4, 5, 6} 10. pej A = {3}, lej meb®e A ®es meJe& μekeÌ³e GHemeb®e

efuene. GkeÀue: DeeefCe {3}

11. pej U = {1, 2, 3, 4} DeeefCe X = {2, 4}, lej X μeesOee.

GkeÀue: efouesues, U = {1, 2, 3, 4} DeeefCe X = {2, 4}

X = {1, 3} mejeJeemeeþer DeefOekeÀ GoenjCes DeY³eeme 1.1 Jej DeeOeeefjle: 1. Keeueerue meb®e ³eeoer He×leerves efuene. i. A = {x|x ner cetU mebK³ee Deens, peer 30 ®eer

efJeYeepekeÀ Deens.}

ii. B = {x|x ner mece vewmeefie&keÀ mebK³ee Deens.}

iii. C = {x|x ne HetCee¥keÀ Deens DeeefCe x2 < 5}

iv. F = {x|x ns ‘LITTLE’ nîee μeyoeleerue De#ej Deens.}

v. E = {x|x W, x N} vi. D = {x|x ner mebK³ee 81 ®es Jeie&cetU Deens.} 2. Keeueerue meb®e iegCeOece& He×leerves efuene. i. A = {2, 4, 6, 8, 10, 12, 14} ii. B = {5, 10, 15, 20, ….} iii. C = {7, 72, 73, 74} iv. D = {51, 53, 55, 57, 59} v. E = {2, 3, 5, 7, 11, 13, 17, 19} DeY³eeme 1.2 Jej DeeOeeefjle: 1. KeeueerueHewkeÀer keÀesCeles meb®e `SkeÀ IeìkeÀ meb®e' efkebÀJee èfjkeÌle

meb®e' Deensle ns meebiee. i. A = {x|x 5 = 0} ii. B = {y|y ner mece cetU mebK³ee 2 Hes#ee ceesþer Deens}

iii. D = {x|x N DeeefCe 3x 1 = 0}

iv. E = {x|x I, x ner Oeve mebK³eener veener DeeefCe $eÝCe mebK³eener veener}

v. C = {x|x N DeeefCe x < 7 DeeefCe x > 11} 2. Keeueerue meb®eeb®es `meeble' efkebÀJee `Deveble' meb®eeceO³es JeieeakeÀjCe

keÀje. i. A = {x|x ner 1 ®eer iegCekeÀ Deens}

ii. C = {x|x ne js<esJej®ee efyebot Deens}

iii. D = {1, 2, 3, 4, …., 100} iv. E = {x|x N DeeefCe x ner efJe<ece mebK³ee Deens}

19


DeY³eeme 1.3 Jej DeeOeeefjle: 1. Keeueerue meb®eeleerue GHemeb®eeb®es mebyebOe oeKeJee. P = veeieHetjceOeerue meJe& jefnJeeμeeb®ee meb®e X = ye[esoeceOeerue meJe& jefnJeeμeeb®ee meb®e Y = ceneje<ìêceOeerue meJe& jefnJeeμeeb®ee meb®e T = iegpejeleceOeerue meJe& jefnJeeμeeb®ee meb®e 2. Keeueerue meb®eebceOeerue `GHemeb®e mebyebOe' Jesve Deeke=Àleer®³ee

meene³³eevess oeKeJee. A = {2 , 8} B = {x|x = 2n, n 4 DeeefCe n N} C = {x|x mece vewmeefie&keÀ mebK³ee 20} 3. pej A = {x, y} Demesue lej, meb®e A ®es meJe& μekeÌ³e GHemeb®e

efuene. 4. mel³e keÀer Demel³e les meebiee: i. ne mJele:®ee SkeÀ GHemeb®e Deens. ii. pej A B DeeefCe B A, lej A = B iii. efjkeÌle meb®e ne meJe& meb®eeb®ee GHemeb®e Demelees. DeY³eeme 1.4 Jej DeeOeeefjle: 1. Keeueerue Òel³eskeÀ GoenjCeeleerue meb®eeb®es `meb³eesie meb®e' efuene. i. A = {5, 15, 25}, B = {10, 20, 30} ii. H = {3, 6, 9, 12, 15} , F = {3, 4, 5, 6} iii. M = {x|x N DeeefCe x ne 12 ®ee ogYeepekeÀ Deens} N = {x|x N DeeefCe x ne 12 ®ee cetU ogYeepekeÀ Deens} 2. Keeueerue Òel³eskeÀ GoenjCeeleerue meb®eeb®es `íso meb®e' efuene. i. A = {5, 6, 7}, B = {8, 9, 10} ii. M = {10, 20, 30, 40, 50}, N = {20, 40, 60} iii. N ne vewmeefie&keÀ mebK³eeb®ee meb®e Deens DeeefCe W ne

HetCe& mebK³eeb®ee meb®e Deens. iv. P = {a, b, p, d, q}, R = {q, r, s, p} 3. pej U = {x|x ner vewmeefie&keÀ mebK³ee 15 Hes#ee ueneve Deens}

ne efJeμJemeb®e Deens. A = {1, 3, 4, 5, 9},

B = {3, 5, 7, 9, 12} lej He[leeUtve Hene: (A B) = A B

4. U = {x|x I DeeefCe 3 x 3},

A = {2, 0, 2}, B = {0, 1, 2, 3} μeesOee i. A

ii. B iii. (A B) iv. A B DeY³eeme 1.5 Jej DeeOeeefjle: 1. Keeueerue Deeke=Àleer®³ee meene³³eeves Heg{erue meb®e efuene: i. A ii. B iii. U iv. A B v. A B 2. mecepee, A DeeefCe B oesve Demes meb®e Deensle p³eele,

n (A) = 17, n (B) = 23, n (A B) = 38. lej n (A B) μeesOee.

3. μeeUsleerue 240 cegueeb®³ee cegueeKeleer IesTve l³eeb®es íbo

efJe®eejues iesues. 150 cegueebvee Heesmìe®eer efleefkeÀìs pecee keÀjC³ee®ee íbo neslee. 80 cegueebvee HegmlekesÀ Jee®eC³ee®ee íbo neslee. 40

cegueebvee ³eeHewkeÀer keÀener®e DeeJe[le veJnles. p³ee cegueebvee Heesmìe®eer efleefkeÀìs pecee keÀjCes DeeefCe HegmlekesÀ Jee®eCes ³ee oesvner iees<ìer DeeJe[leele Deμee cegueeb®eer mebK³ee efkeÀleer Deens?

4. 50 efJeÐeeL³ee¥®³ee Jeiee&le, 35 efJeÐeeL³ee¥vee YeeweflekeÀ μeem$e

DeeJe[les, 30 efJeÐeeL³ee¥vee ieefCele DeeJe[les DeeefCe 3

efJeÐeeL³ee¥vee keÀener®e DeeJe[le veener. ³eeleerue efkeÀleer peCeebvee oesvner efJe<e³e DeeJe[leele DeeefCe efkeÀleer peCeebvee HeÀkeÌle YeeweflekeÀμeem$e DeeJe[les?

5. Jejerue Deeke=ÀleerJeªve μeesOee : i. A B ii. n (A B) iii. (A B) iv. n (A B) v. A B

A B

18126

3

159

U

24

A B

7

105

1

49

U

311

6

2 8

12

20

F. 9 Jeer : yeerpeieefCele yengHe³ee&³eer ÒeMve 1. pej B = {x|x ne Fbûepeer JeCe&ceeuesleerue mJej Deens}, lej B

ne meb®e ³eeoer He×leerves Heg{erue ÒeceeCes efueefnuee peeF&ue. (A) {a, e, i, u} (B) {a, e, p, o} (C) {a, e, c, d} (D) {a, e, i, o, u} 2. KeeueerueHewkeÀer keÀesCelee meb®e `Deveble meb®e' veener?

(A) N (B) W (C) I (D) ³eebHewkeÀer keÀesCeleener veener. 3. A = {z|z + 6 = 0} ne _______ Deens. (A) efjkeÌle meb®e (B) SkeÀ IeìkeÀ meb®e (C) Deveble meb®e (D) meeble meb®e 4. efjkeÌle meb®e ne Òel³eskeÀ meb®ee®ee _______ Demelees. (A) GHemeb®e (B) ³egkeÌle GHemeb®e (C) Deeflemeb®e (D) efJeμJemeb®e 5. pej A = {x|x ne legce®³ee keÀe³ee&ue³eeleerue efJeYeeie I

ceOeerue keÀce&®eejer Deens.}

B = {y|y ne legce®³ee keÀe³ee&ue³eeleerue efJeYeeie II

ceOeerue keÀce&®eejer Deens.}

DeeefCe C = {z|z ne legce®³ee keÀe³ee&ue³eeleerue keÀce&®eejer Deens.}, lej

(A) C A (B) A B (C) A C (D) C B 6. pej A B DeeefCe B A, lej A DeeefCe B ns

_______ meb®e Deensle. (A) meceeve (B) efJeefYeVe (C) Deefle (D) efJeμJe 7. pej A P, B P, lej (A B) ________.

(A) A (B) B (C) P (D) 8. pej A = {3, 4, 7, 8, 9} DeeefCe B = {7, 8, 10, 11}

lej A B = ?

(A) {3, 4} (B) {7, 8} (C) {10, 11} (D) {10} 9. A DeeefCe B ³ee keÀesCel³eener oesve meb®eebmeeþer, A B = ?

(A) {x|x B efkebÀJee x A}

(B) {x|x A efkebÀJee x B}

(C) {x|x A DeeefCe x B}

(D) {x|x A DeeefCe x B}

10. pej U = {1, 2, 3, 4, ….} DeeefCe A = {2, 4, 6, 8, ….} lej A = ?

(A) {2, 4, 6, ….} (B) {1, 3, 5, 7, ….} (C) {0, 1, 3, 5, ….} (D) {0, 2, 4, 6, 8, ….} 11. pej U = {4, 5, 6, 7, 8, 9}, P = {5, 6, 7, 8},

Q = {4, 6, 8, 9} lej, P Q = ?

(A) {4, 5, 7, 8, 9} (B) {4, 5, 7, 9} (C) {6, 7, 8} (D) {4, 6, 7, 8, 9} 12. Heg{erue Jesve Deeke=ÀleerceO³es, pej n(P Q) =70, lej x = ?

(A) 5 (B) 3 (C) 6 (D) 8 13. pej n(A) = 10, n(B) = 25 DeeefCe n(A B) = 15,

lej n(A B) = ?

(A) 20 (B) 0 (C) 10 (D) 5 mejeJeemeeþer DeefOekeÀ GoenjCeeb®eer GÊejs DeY³eeme 1.1 Jej DeeOeeefjle: 1. i. A = {2, 3, 5} ii. B = {2, 4, 6, 8, ….}

iii. C = {2, 1, 0, 1, 2} iv. F = {L, I, T, E} v. E = {0}

vi. D = {9, 9} 2. i. A = {x|x = 2n, n N DeeefCe n < 8}

ii. B = {x|x = 5n DeeefCe n N}

iii. C = {x|x = 7n, 1 n 4}

iv. D = {x|x N, x efJe<ece HetCee¥keÀ Deens DeeefCe 50 < x < 60}

v. E = {x|x ner cetU mebK³ee Deens DeeefCe 1 < x < 20} DeY³eeme 1.2 Jej DeeOeeefjle : 1. A, E ns SkeÀ IeìkeÀ meb®e Deensle. B, D, C ns efjkeÌle meb®e Deensle. 2. D ne meeble meb®e Deens. A, C, E ns Deveble meb®e Deensle.

P Q

40 x 35 x x

21


DeY³eeme 1.3 Jej DeeOeeefjle: 1. P Y, X T 2. A = {2, 8}, B = {2, 4, 8, 16} C = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} 3. , {x}, {y}, {x, y} 4. i. mel³e

ii. mel³e

iii. mel³e DeY³eeme 1.4 Jej DeeOeeefjle: 1. i. A B = {5, 10, 15, 20, 25, 30} ii. H F = {3, 4, 5, 6, 9, 12, 15} iii. M N = {1, 2, 3, 4, 6, 12} 2. i. A B = ii. M N = {20, 40} iii. N W = {1, 2, 3, ….} iv. P R = {p, q} 4. i. A = {3, 1, 1, 3} ii. B = {3, 2, 1} iii. (A B) = {3, 1} iv. A B= {3, 1} DeY³eeme 1.5 Jej DeeOeeefjle: 1. i. A = {2, 3, 6, 7, 8, 11, 12} ii. B = {1, 3, 4, 6, 9, 11, 12} iii. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} iv. A B = {1, 2, 4, 5, 7, 8, 9, 10} v. A B = {5, 10} 2. 2 3. 30 4. 18 efJeÐeeL³ee¥vee oesvner efJe<e³e DeeJe[leele DeeefCe 17

efJeÐeeL³ee¥vee HeÀkeÌle YeeweflekeÀμeem$e ne efJe<e³e DeeJe[lees. 5. i. A B = {6, 12} ii. n(A B) = 7

iii. (A B) = {3, 9, 15, 18, 24} iv. n(A B) = 2 v. A B = {3, 9, 15, 18, 24}

yengHe³ee&³eer Òeμveeb®eer GÊejs 1. (D) 2. (D) 3. (B) 4. (A) 5. (C) 6. (A) 7. (C) 8. (B) 9. (B) 10. (B) 11. (B) 12. (A) 13. (A)

A

B

2

C

8

10

12

20 1418

6

4 16

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