कक्षा ८ गणित

kf7

10 ;dx" (Sets)

10.0. kg' /jnfs] g (Review)

tnsf ;dx" x¿sf] cWoog u/L lbOPsf kZ| gx¿sf pQ/x¿sf] vfh] L u/ M

U = { 1 bl] v 20 ;Dddf ;ªV\ ofx¿ }

A= { 20 eGbf ;fgf 3 sf ckjTox{ ¿ }, / B= { 20 eGbf ;fgf 4 sf ckjTox{ ¿} eP

-s_ ;dx" x¿ U, A, / B sf ;b:ox¿ ;r" Ls/0f u/ .

-v_ ;dx" x¿ U, A, / B nfO{ eg] lrqdf k:| tt' u/ .

-u_ eg] lrqsf] ko| fu] u/L tnsf ;dx" x¿ kQf nufpm M

(i) A∪B (ii) A∩B

-3_ A / B ;dx" U sf s:tf ;dx" x¿ xg' \ <

-ª_ ;dx" A df kg{] t/ ;dx" B df gkg{] ;b:ox¿sf] ;dx" lgdf0{ f u/ .

-r_ ;dx" U df kg{] t/ A ∪ B df gkg{] ;b:ox¿sf] ;dx" kQf nufpm .

dflysf kZ| gx¿sf af/d] f xfdLn] sIff 7 df g} kl9;ss] f 5f“} . ca xfdL ;dx" x¿sf] km/s
/ ;dx" sf k/" ssf af/d] f cWoog ub5{ f“} .

10.1. ;dx" x¿sf] km/s ( Difference of Sets)

tnsf eg] lrqx¿ cWoog u/ / s] s] bl] vG5, ;an} ] cfcfkm\ gf sfkLdf nv] M

U U U

AB Aa B e 4
bd
l;tf cfOt 8f]Ndf cf B 1 A
xl/ Zofd 5 2 6
3

;a} eg] lrqdf ;dx" B afxs] A sf] efu dfq 5fof kfl/Psf] 5 . ;dx" A sf] dfq efudf 5fof
kfl/Psf] 5 . B sf] sg' } klg efudf 5fof kfl/Psf] 5g} / 5fof kf/s] f] efun] ;dx" A df kg{] t/
B df gkg{] ;b:ox¿sf] ;dx" nfO{ hgfp5“ . of] g} ;dx" A af6 ;dx" B sf] km/s xf] .

xfdf| ] ul0ft, sIff * 83

olb ;dx" A / ;dx" B ;jJ{ ofks ;dx" U sf pk;dx" x¿ xg' \ eg] ;dx" A df kg{] t/ ;dx" B df
gkg{] ;b:ox¿sf] ;dx" nfO{ jf ;dx" A df dfq kg{] ;b:ox¿sf] ;dx" nfO{ A km/s B elgG5 /
o;nfO{ A-B n] hgfOG5 . A-B = {x:x ∈ A / x ∉ B}

dflysf] klxnf] eg] lrqdf A = { ;Ltf, xl/, cfOt } 5 / B= {cfOt, 8fN] df, Zofd} 5 .

A df dfq kg{] ;b:ox¿ A–B = { ;Ltf, xl/, cfOt } – {cfOt, 8fN] df, Zofd}
= { ;Ltf, xl/ } eof] .

To:t} bf;] f| df A -B= {a, b, c} / t;] f| df A–B= {4, 5, 6} eof] -s;/L <_

pbfx/0f 1

olb, U = {cfbz{ df= lj=sf ;Dk0" f{ ljBfyLx{ ¿sf] ;dx" }

A = { cfbz{ df=lj=sf sIff 8 sf ljBfyLx{ ¿sf] ;dx" } /

B = { cfbz{ df=lj=sf ;Dk0" f{ 5fqfx¿sf] ;dx" } eP A-B / B-A kQf nufpm / eg] lrqdf
5fof kf//] bv] fpm .

;dfwfg A U
oxf,“ U= {cfbz{ df= lj=sf ;Dk0" f{ ljBfyLx{ ¿sf] ;dx" } A–B B

A = { cfbz{ df=lj=sf sIff 8 sf ljBfyLx{ ¿sf] ;dx" }

B = { cfbz{ df=lj=sf ;Dk0" f{ 5fqfx¿sf] ;dx" }

ca, A–B = { x: cfbz{ df=lj=sf sIff 8 sf ljBfyLx{ ¿ xg' \ t/ 5fqf xfO] gg\ )}

A–B ={ cfbz{ df=lj=sf sIff 8 sf 5fqx¿sf] ;dx" } U B
A

km/] L, B–A = {x: cfbz{ df=lj=sf 5fqf t/ sIff 8 sf xfO] g} B–A

= { cfbz{ df=lj=sf sIff 8 sfafxs] sf 5fqfx¿sf] ;dx" }

gf6] M 1. A-B ≠ B-A
2. A– B ∪ B – A ePdf o;nfO{ ;dldtLo km/s (Symmetrical Difference) elgG5 .

pbfx/0f 2

olb U = {a, b, c, d, e, f, g, h, i, o, u} , A= {a, b, c, d, e}, B = {a, e, i,o, u} /
C = { d, e, f, i, } eP tnsf ;dx" x¿ kQf nufpm / eg] lrqdf k:| tt' u/ M

-s_ A-B -v_ B-C -u_ A∪(B-C)

-3_ U-(A∪B) -ª_ (A∪B) - (A ∩B) -r_ (A-B) ∪ (B-A)

84 xfdf| ] ul0ft, sIff *

;dfwfg Ab a BU
oxf,“ U = { a, b, c, d, e, f, g, h, i, o, u} , A = { a, b, c, d, e } c e
d i
B = { a, e, i,o, u } / C = { d, e, f, i } o
-s_ A– B = {x:x ∈ A / x ∉ B} u

= {a, b, c, d, e} - {a, e, i, o, u} A-B
= {b,c, d}
BU
-v_ B-C = {x: x ∈ B / x ∉ C} a

= { a, e, i,o, u } - { d, e, f, i, } o e d
= {a, o, u} u f
C
-u_ A∪(B-C) = {x∈A cyjf x ∈ B-C}
B-C
= {a, b, c, d, e} ∪ {a, o, u}
= {a, b, c, d, e,o,u } A Bb- C a U
c e oB
d u

C
AU

B

-3_ U-(A∪B) = {x ∈U / x∉A∪B} C

oxf,“ A∪B = { a, b, c, d, e} ∪ { a, b, c, d, e, i,o, u } A U-(A∪B)
= { a, b, c, d, e, i, o, u} xG' 5 . BU

= { a, b, c, d, e, f, g, h, i, o, u} - { a, b, c, d, e, i, o, u }

= {f,g, h}

-ª_ A ∩ B = { a, b, c, d, e} ∩ {a, e, i, o, u} = {a, e} 5fof kfl/Psf] efu
(A∪B) - (A ∩ B)]
t;y,{ (A∪B) - (A ∩ B) = {a, b, c, d, e, i, o, u} - {a, e}
1111111111111111222222222222222233333333333333334444444444444444A55555555555555556666666666666666777777777777777788888888888888881111111111111111999999999999999922222222222222220000000000000000333333333333333311111111111111114444444444444444222222222222222255555555555555553333333333333333666666666666666644444444444444447777777777777777555555555555555588888888888888886666666666666666999999999999999900000000000000001111111111111111B222222222222222233333333333333334444444444444444555555555555555566666666666666667777777777777777 U
= {b, c, d, i, o, u} (A−B) ∪ (B − A)

-r_ oxf,“ B-A = { x ∈ B / x ∉ A} 85

= {a, e, i, o, u } - {a, b, c, d, e }
= {i, o, u }

/ -s_ af6 xfdLnfO{ yfxf 5 A-B = { b, c, d}
ca, (A-B) ∪(B-A) = { b, c, d} ∪ {i, o, u }

= {b, c, d, i, o, u}

-ª_ / -r_ af6 s] kfof} < nv] .

xfdf| ] ul0ft, sIff *

cEof; 10.1

1. olb A = {2, 4, 6, 8,10} / B = { 4, 8, 10, 12, 14} eP tnsf ;dx" x¿ kQf nufpm M

-s_ A∪B -v_ A∩B -u_ A-B

-3_ B-A -ª_ (A-B)∪(B-A)

2. kZ| g g= 1 sf ;dx" x¿nfO{ 56' 6\ f56' 6\ } eg] lrqdf k:| tt' u/ .

3. olb P = {20 eGbf ;fgf hf/] k0" f{ ;ªV\ ofx¿sf] ;dx" } / Q={20 eGbf ;fgf 4 sf ckjTox{ ¿sf]
;dx" } eP P-Q / Q-P nfO{ eg] lrqdf k:| tt' u/ .

4. lbOPsf] lrqdf tnsf ;dx" x¿nfO{ 56' 6\ f56' 6\ } 5fof kf//] bv] fpm .

-s_ A-B -v_ B-C A BU

-u_ C-A -3_ (A∪B)-C

-ª_ A-(B∪C) -r_ (B∩C)-(C∩A) C

5. olb U={1,2,3,4,5,6,7,8,9}, A={2,4,6,8}, B={3,5,7,9} / C={3,6,9} eP

-s_ (A∪B)-C -v_ (B∪C)-A -u_ (C∪A)-B kQf nufO{ eg] lrqdf k:| tt' u/ .

6. tnsf eg] lrqx¿sf] 5fof kf/s] f] efusf] gfd ;ªs\ t] df nv] M
P QU A BU
AB U

n

C

7. olb A = {a, e, i, o, u } , B = {i, o, u,w} / C = {e, i, o} eP (A-B)-C / A-(B-C) nfO{
56' 6\ f56' 6\ } eg] lrqdf k:| tt' u/ .

8. U = { ltdf| ] ljBfnodf sIff 8 sf ;Dk0" f{ ljBfyLx{ ¿sf] ;dx" }
A = { sIff 8 sf skbL{ vN] g dg k/fpgs] f] ;dx" }
B = { sIff 8 sf 8G8Llaof] vN] g dg k/fpgs] f] ;dx" }
C = { sIff 8 sf km' 6an vN] g dg k/fpgs] f] ;dx" }
olb A, B / C ;a} kl| tR5l] bt ;dx" x¿ eP A, B, C sf] ;DaGwnfO{ egl] rqdf k:| tt' u/ .

9. kZ| g 8 sf cfwf/df tnsf ;dx" x¿nfO{ 5fof kf//] bv] fpm M xfdf| ] ul0ft, sIff *
-s_ skbL{ dg k/fpg] ljBfyLx{ ¿sf] ;dx"
-v_ skbL{ / 8G8Llaof] dg k/fpg] ljBfyLs{ f] ;dx"
-u_ skbL,{ 8G8Llaof] jf km' 6an vN] g dg k/fpgs] f] ;dx"
-3_ tLg cf6] } vn] vN] g dg k/fpgs] f] ;dx"

86

U
A

1A 11 U

3 2 4 13
6 8 15
17
5 10 12 19

7 14 16
18

9A

xfdf| ] ul0ft, sIff * 87

88 xfdf| ] ul0ft, sIff *

xfdf| ] ul0ft, sIff * 89

10.3 eg] lrqsf] ko| fu] ( Use of Venn Diagrams)

ul0ft1 John Venn n] la;f“} ztfAbLdf ;dx" sf ljm| ofx¿nfO{ lrqåf/f k:| tt' u/s] f lyP . pg}
ul0ft1sf] gfdaf6 eg] lrq (Venn- Diagram) elgPsf] xf] . ;dx" sf zflAbs ;d:ofx¿nfO{
eg] lrqsf] ko| fu] u/L ;dfwfg ug{ ;lsG5 .

tnsf] pbfx/0f cWoog u/f“} M
u08sL c~rnsf lhNnfx¿sf] ;dx" = { sf:sL, :ofªh\ f, tgx,'“ ndh'ª, uf/] vf, dgfª }

wfn} flul/ c~rnsf lhNnfx¿sf] ;dx" = {jfUnª' , DofUbL, kjt{ , d:' tfª}

u08sL c~rnsf lhNnfx¿sf] ;dx" nfO{ G dfGbf, G = { sf:sL, :ofªh\ f, tgx,“' ndhª' , uf/] vf, dgfª} .
u08sL c~rnsf 6 cf]6f lhNnfx¿ ;d"x G sf ;b:ox¿ 5g\ . o;nfO{ G ;d"xsf]
u0fgfTdstf elgG5 . To:t,} wfn} flul/ c~rnsf lhNnfx¿sf] ;dx" D sf] u0fgfTdstf 4 eof] .

sg' } klg ;dx" df ePsf] hDdf ;b:ox¿sf] ;ªV\ ofnfO{ pSt ;dx" sf] u0fgfTdstf -cardinality_ elgG5 .

oxf“ M dflysf] ;dx" D sf] u0fgfTdstf 4 5 . o;nfO{ n(D) = 4 nl] vG5 .

olb, U ={ ;fs{ /fi6x« ¿sf] ;dx" } A BU11111111111111222222222222223333333333333344444444444444555555555555556666666666666677777777777777888888888888889999999999999900000000000000111111111111112222222222222233333333333333111111111111112222222222222233333333333333444444444444445555555555555566666666666666777777777777778888888888888899999999999999000000000000001111111111111122222222222222
A ={ gk] fn, ef/t, kfls:tfg }
B = { e6' fg, aªu\ fnfbz] , >Lnªs\ f, dflNbE;, ckmuflg:yfg } eP,

A∪B = { gk] fn, ef/t, kfls:tfg, e6' fg, aªu\ fnfbz] , >Lnªs\ f, ckmuflg:yfg } xG' 5 . n(A∪B) slt xG' 5 <

oxf,“ A∪B df hDdf 8 cf6] f ;b:ox¿ 5g\ . To;sf/0f n(A∪B) = 8 eof] .

kml] /, n(A) = 3 / n(B) = 5 5 . n(A) + n(B) = 3 + 5 = 8 eof] .

n(A∪B)=n(A) + n(B) eof] .

bO' c{ f6] f ;dx" x¿ cnlUuPsf 5g\ eg] n(A∪B)=n(A) + n(B) xG' 5 . A B U

To:t,} lbOPsf] eg] lrqdf x/] . 1 1111111111111122222222222222333333333333335444444444444444555555555555556666666666666677777777777777 6
2 7
A = {1, 2, 3, 4, 5}; B = { 4, 5, 6, 7 } n(A) = 5, n(B) = 4 5 . 3

km] l/, A∪B = {1, 2, 3, 4, 5,6,7} / A∩B = {4, 5} 5 .
n(A∪B) = 7 / n(A∩B) = 2

To;sf/0f, eg] lrqsf] ko| fu] ubf{ n(A) + n(B) = 9 eof] t/ n(A∪B) = 7 5 .

hg' n(A) + n(B) eGbf 2 jf n(A∩B) n] sd 5 .

To;sf/0f, n(A∪B) = n(A) + n(B) - n(A∩B) eof] .

bO' c{ f6] f ;dxÒ A / B kl| tR5l] bt ;dxÒ x¿ eP n(A∪B) = n(A) + n(B) - n(A∩B) xG' 5 . ;fy,}

;dxÒ A df dfq kg{] ;b:ox¿sf] ;ªV\ of 3 5 . t;y,{ n0(A) = 3 5 .

n0(A) = n(A) - n(A∩B)

;dxÒ B df dfq kg{] ;b:ox¿sf] ;ªV\ of 2 5 . t;y,{ n (B) = 2 5 .
0

n0(B) = n(B) - n(A∩B) xfdf| ] ul0ft, sIff *

90

xfdf| ] ul0ft, sIff * 91

92 xfdf| ] ul0ft, sIff *