SETTHEORY

SINOPSIS

SET: Astisawell detinad eolletion ofobjcts. That is ioragiven ohjct. itisposible to detenine. whether
that object belongs to the given collection or not.

Notations: The sets are sally denoted by capital letters A. B.(. ele. and the nembers or elements of thesct
are denoted by lower-case letters a. b, e. cte.If'x is a member ofthe set A,we write e A(read as x
belongs to A) and if xis not amember ofthe set A, wewrite xeA (read as xdocs not belong to A). If
xand y both belong toA. we wrie x.y e A.

Representatation ofa Set:

Isually. sets are represented in the following two ways.
1. Roster form or Tabular form.
2. Set Builder fom or Rule Method.

Roster Form:
In this form, we list allthe members of the setwithin braces(curly brackets)and separate these by
commas.

For exanmple. the set Aofallodd natural numbers less than l10 in the roster fom is written as:
A=1,3. S.7.9}
Note :
1) In roster form. every clement ofthe set is listed only once.
2)The order in which the elements are listed is immaterial.
For example. each ofthe following scts denotesthe same set 1.2. 3}. (3.2. 1.1.3.2;.

Set - Builder Form :

In this form.we write a variable (say x) representing any member of the set followed by a property
satisfied bycach member of the set.
For example. the set.Aofall prime numbers less than 10 in the set-builder form is written as
A-(xx is a prime number less than 10}

Types of Sets:
Empty Setor Null Set:
Aset which has no elenment is called the null set or emply set. It is denoled by the smbol .
Torexa1ple, cach ofthe following is anull set :
1. The setof allreal numbers whose square is -1.
 set.
denoted
intin1te is B A proDer
andit every of of
equal. suDerset
subset orB
an integers. A
and ofB of'
called set number
B
not a
the called
called subset
in are
is all of ofAis the
it of number but is is proper
ise. set A B and
Otherw element cqual.cquivalent thenA).
the a gm
cardinal
set. set.
whereas. n(B).be B. contains IS
:A ale
0. set. infinile ifevery not are of
non-empty is elenent as A
odd. member finite the = nced (TCad of
ifn(A) 8.9} B subsets
sct an called
A-B, Bor
2. and set. a sct. sets B={1.6. BTA
is evena calledfiniteis an in
square called singleton only
a
integer},
finite is
setA as equivalent.
cquivalent is
ofAcontained itsel[. of
both is is written OT ¢B. number
whose week isa
elements and AzB set.
ofevery
subset
hose is finite element of
are element called an element. A
supersct
w that set. a is a equal, be but 3.2} is write the
in in to equivalent A wrile
numbers
inteuers
is singleton of days (x|x elements be said A={4.5, Ifcvery as
we
then
one clement number no (read anda
all or has to are andAB.We
B., m.
atleast said sets.BDA of thenA)
rational of ..... which B are sets subset
those one a finite distinct subset
set are and
of only
is
(0} Set: 1,2. setsFor
example,
the
two is with -MATHEM
the B A cBor set -1:
all ofallconsisting has -1.0. ¢ of and setsequal be ofA).
superset
a
is a empty
set
of having example.Infiniteexample. set Cardinal
Number:
number :Sets
Equivalent B AcB set not
DIf 2M
which A finite and g
set set Singleton
Set: empty
(.....2. n(A). sets A. Clearly, A Everyis ie are
The The set :Sets
Equal The
3)IfA Ey1eA
set For and set The Twoin is Two A write ofA
For An by Let
Subset: 2) 4)
2. 3. A A A B :Notes PUC
Finite
|-
 Power Set:
Ihe set of all subsets ola given set Ais called the power sct ofA and is denoted by PA).
For example. il'A {1.2. 3}.then
PA)-{0.(|23:.1.2}. (1.3}. (2.3}. |1.2. 3}
Clearly. ifA has nelements. thenits power set PA)contains exactly g clements.

Operations on Sets :
Union of Two Sets:
The union of twosets Aand B. written as AJB (read as Aunivon B). is the set consisting of allth
clementswhich are either in Aor in Borin both.
Thus, AUB=x:xE Aor xe B}Clearly. xe AUB»e Aor xeB.
and x AUB’¢A and x B.
For example. ifA-(a. b. c. d} and B=lc.d. e. f}.then AB a. b.c. d. e.f9.
Intersection of TwoSets :
The intersection oftwo sets Aand B. written as AOB (read as AinterSection B) is the sct consisting
all the commonelements ofA and B.
Thus, AOB=}x:NEA and xe BClcarly. XE ABxeA and xe B
XAB’x AOr X B.
For example, ifA=a. b. c. d} and B-(c. d. c. } then ABc. d}.

Disjoint Sets :
Two setsA and Bare said to be disjoint. if AnB =0.i.e..A and B have no clement in common.
For example. ifA-1.2, 5} and B=(4.6}. then A nB=0.so Aand Bare disjoint sets.

Difference of Two Sets :

IfA andBare two sets, then their dilference A- BorA\B(or) is defined as:
B
A-B=x:xEA and x B
Similarly. B-A=x:xeB and xEA
For example. ifA={1,2, 3,4} and B-{1. 3. 5.7.9}. then A-B-2.4} and B-A-7.9:.

Important Results :
1. A-BB-A
2. The sets A-B, B-A and A OBare disjoint sets.
3. A- BcA and B-AcB
4. A- = Aand A-A =

I- PUC MATHEMATICS -
Symmetric Difference of"Two sets:
The symmetricdifference oftwo sets Aand B. denoted by AAB. 0s defincd as
AAB =(A - B)U(B- A).

For example. if A-{1.2,3.4.5} and B-1.3.5.7.9; then

AAB =(A-B)U(B-A)
-{2,4}{7.9}
=(2.4,7,9}
Complement of a Set :
IUisa universal set and Ais a subset ofU, then the complement ofA is the set which
contains thos
elements ofU, which are not contained in Aand is denoted by A' or
A =x:xeUand xe A} A.Thus,
For example, ifU={1,2. 3.4, .... and A=(2. 4.
6.8..... then. A={1.3.5.7. ....
Important Results :
a) U'=¢ b) '= U c) AUA'=U d) AnA'=
ALGEBRA OF SETS:
Idempotent Laws :
For any set A, we have
a) AUA =A b) AnA =A

Identity Laws:
For any set A, Uis universal set, we have
a) Au = A b) An¢= c) AUU=U d) AnU =A
Commutative Laws:
For any two sets Aand B. we have
a) AUB= BUA b) AnB= BAA

Associative Laws :
For any three setsA. B andC, we have
a) Aw(BUC)=(AuB)UC
b) An(BnC) =(ArB)nC

|- PUC MATHEMATICS -
Distributive Laws:
For any three sets A. B andC, we have
a) A(BoC)=(AUB)n(AuC)
b) An(BC)=(AnB)U(An)
For any two sets Aand B, we have
a) P(A)^P(B) =P(AB)
b) P(A)P(B)cP(AUB),where P(A) is the power set of A.
IfA is any set. (A-A
Demorgan's Law's for any three sets A. Band C
a) (AUB) =AoB b) (AB) =AuB
c) A-(BUC)=(A-B)^(A-C) d) A-(BoC)=(A-B)U(A-C)

Key Results on Operations on Sets

AcAUB. BeAB. AnBcA. AnBeB

A-B= AnB

(A-B)UB= AUB

(A-B)^B= ¢

AcB BeA

A-B=B -A

(AUB)n{AB)=A
AUB=(A-B)u(B-A)U(AnB)

A-(A-B) =AnB
A-B= B-A A =B

AUB=AnBA=B
An(BAC)=(AnB)A(AnC)
|- PUC MATHEMATICS
PRACTICE QUESTIONS

1) -reR: x+2r-1
4r +31-E R equals to

1) R-{0} 2) R-0, 1,3} 3) R-{0.-1,-3} 4)

2) IfSisa set with 10elements and 4 =i(x, ):x,ye S,x*y} then the number of clementsin Ais
1) 100 2) 90 3)50 4) 45

3) Ina certain town 25°% families owna cell phone, 15% fanmilies own ascooter and 65% families own
neither acell phone nor ascooter. If 1500 families own both a celiphone and ascooter then the total
number of families in the town is
1) 10000 2) 20000 3) 30000 4) 40000

4) If 4 c, cA, c....cAf nd n( A) =i-3 then

1)5 2) 7 3) 9 4) 11

5) Let S=(x:x is a positivemultiple of3 less than 100}

P={x:xis a prime number less than 20}. Then n(S) +n(P) =
1) 34 2) 41 3)33 4) 30

6) IfA={1,3,5,7,9,11.13,15,17} ,B=(2.4. 18} and Nthe set of natural numbers is the universal
set, then 4'u[(4UB)^B]
1) ¢ 2) N 3)A 4) B

7) If sets Aand Bare defined as A

=(.):y=,0zxeR B={(,):y= -X,re R} then
1) 4B= A 2) AnB= B 3) An B= 4) AUB= A

8 Two finite sets have mand nelenments respectively. The total number of subsets of first set 56 more
than the total number of subsets of thesecond set. The values of mand n respectively are
1) 7.6 2) 5.1 3)6,3 4) 8,7

9) Ifsets Aand Bare defined as 4 =(x,y):y =e',xe R}B = (x. ):y=x,xeR} then
1) BcA 2) Ac B 3) AnB= 4) AUB = A

10) The smallest set Asuclh that AI, 2} ={1.2,3,5,9; is

1) (2,3,5} 2){3.5.9} 3){1.2,5,9} 4){1.2}

I-PUCMATHEMATICS
 IM n(U)=700, (A) = 200. B)- 300 lAoB)= 100 then n1o B')) Is eqLlal t0
I) 400 2) 240 3) 300 4)500

12) If aN = fax:xe N} then the set

1) 21N
3NO7N is
2) 1ON 3) 3N 4) 20N

13) The set (ABYU(BoC)=

1) AUBUC 2)A'U B 4) AB
3) AJ"

14) If A=:eZ.xs4 then the number of subsets having at least two elements is
1) 511 2) 503 3) 502 4) 501

15) Let X =..):x,p,zeN,xty+z =10.r<y< and

Y=(x,y.=):x. ,zeN.y =x- then X o =
) (2,3,5)} 2) (1.4.5)} 3) (5.1.4)} 4)}(1, 4,5).(2,3.5)}
16) In survey shows that 63% ofAmericans like oranges where as 76% like apples. Ifx% ofAmericans
like both then
1) x=39 2) x=63 3) 39 <r<63 4) 39 <xs63
17) IfP(A) denotes the power set ofA and A = then what is the number of elements in
P{P{P{P(A)}}}
1)0 2) 16 3) 1 4) 4

18) Let d=<xeR:<},B={xeR:-z} and AUB - R-D hen D=

){x:l<xs2} 2) (x:1sx<2} 3) (x:1<xs2} 4)none of these

19) If n(4) =i+l and A, cA,c A, c...c Ag then 2

1) 100 2)99 3) 98 4) 101

20) Let A={1, 2. 3,4}and B= (2.3, 4, 5.6} then 4AB 0s equal to

1) (2, 3, 4} 2) {1} 3) {5,6} 4) (1. 5,6}
21) In aclass of 55 students, the number of students studying different subjects are 23 in
mathematics 24
in physics, 19 in chemistry. 12 in mathematics and physics. 9inmathematics and chemistry 7 in
nhisics andchemistry and 4 in all subjects. The number of students who have taken exactly one
subject is
2)9 3)7 4) all ofthese
)6

|-PUCMATHEMATICS-,.
 22) A. B. Care the sets of letters to spell the words STUDENTS. PROGRESS andCONGRUENT
respectively. Then n(4U(BoO)
1) 8 2)9 3) 20 4) 30

23) Which of thefollowing is empty set

) x:xis rèal number and x -1= 0 2) x:x is real number and x +1 0
3) :xis real number and x° -9= 0 4)x:xis real number and.r' =x+2|
24) Ina city 20 percent of population travels by car. 50percent travels by bus and 10 percent by both
car and bus. Then persons travelling by car or bus is
1)80 percent 2) 40 percent 3) 60 percent 4) 70 percent

25) Given n(U)=20. n(A) =12,nB) =9,n(AnB) =4 and n(AUB)

)17 2) 9 3) 11 4) 3

. 26) The set 1,4.9, 16. 25.... can be written as

1) xx=nneN 2) x:=n .neZ
3) *:x=n ,ne N 4)r:r=n.ne N
27) Let S= (x:x is apositive multiple of3 less than 100} P= {x:x isaprime number less than 20} then
n(S)+n(P) is
1) 34 2)41 3) 33 4)30

28) Two finite sets have pand qnumber of elements. The total number ofsubsets of the first set is eight
times the total number of subsetsofthe second set. Find the value ofp-q
1) 2 2) 3 3) 4 4) none of these

29) The number of elements in the power set (x:x is aprime number of lessthan l0}
1) 16 2) 4 3) 15 4) 10

30) IfE={1, 2, 3, 4. 5. 6.7.8,9} the subset of Esatisfying 5+ x> 10 is

1) {5, 6, 7, 8,9} 2) {6,7, 8,9} 3){5, 7, 8,9} 4) (6.7, 8,9, 10}

" 31) Which of thefollowing collections is not aset?

1) The collectionofnatural numbers between 2 and 20
2) The collection of numbers which satisfy the equation ° - 5x +6=0
3) The collection of prime numbers between Iand 100
4) The collection of all beautiful flowers in moodbidre

. 32) The set A=(x:x is a positive prime <10}in a tabular form is

1) (1.2, 3. 5. 7} 2) (1,3,5. 7, 9; 3) (2. 3. 5, 7: 4)none of these

I-PUCMATHEMATICS-.
 43) 42) 41) . 40) 39) 38) 37) 36) 35) 34) 33)
CS.
1Yisafinite
6.2
1)
clements second
4,7 1) Two 1) If 1)
bot0h Ina exactly with Suppose 1)
}
0e{Which 21
1)1A hockey.
Cieket1) ieSTATISTICS,
and Outof 4c1)
BThen
35
1) 1)-B
A IfA 1)A=B II
{30, A= AAB 128 total,
finite the class 3 and andB sn
set
set. 60,90, (2x:xe elements. ofthe 800
in games. of 9B A,A, Bare AUBthen = 64
The sets 60 's are played boys
X.set. following two two letters of
If values have N},B= .. students, then
for Let Then IfUA . in
30 A sets
two m n sets, both
8.4 2) P(X)) 2)
7,4 ofand
is ,....4,o 24school, a
2) 25
2)the equal
3 statement 2)B A- then AnB 2)
= 31
2)such played
finite (3x:re number 25 2) =UB, 0c{} 2) 16
2)basket 2) needed to
denote and m n {15, students
elenents. (AU that 224 A
subsets 30,
to are
n is
all ball
the are of =S thirty nCAUB)=36. the plaved =|| B spell
N},C={5x:
45.... students play true? B)-(AnB)
A. set respectively The and the and
B of cricket sets hockey.
three. cricket,MATHEMATICS`
n(P(A)) all number each
3)
4.4
who each
3)
0,4 subsets 3) xe 35
3) and 3)
15 element c{0} 3)B)(B-A)(4- 3) AAB=|| 3) 20
3) 240 3)The
pla y having is nA 80240 3)
ot (10.
= subsets N} 20 equal numberplaved AAB
n( of 20. then neither
students plaved
ofS to B)= =
PB))+ and X 30.... elements 5 A- and
of is belongs ofboys bothockey h
let the An(BOC) l6 B B
I5n(X) pl ay cricket
first and is
then 7,
4)7 tennis to and and tshetof
4)
0,1 denote set 4) 15
4) 45
4) {0}0c
4) AAB=A- 4)B n(4 who
exactly B 160 4) 4)
n(A) is (7 . 52 did and 336
112 14, and .B, 4) - none
4)none
theseof B)=T. notbasketplayed letters
and the more 21. 10 10,......3,
numberof play of
nB)is .... students of these need
than 4's the ball. spellto
nen basketball
and aren game 0
thatof play nD) playec
sets
TICS 54) 53) 52) 51) 50) 49) 48) 47) 46) 45) 44)

Which 1)
AIfA AnC 1) If four 10
1) Ina clements
D1
Suppose
students.
I)atleast In 8N
1)Iffor
30 {x:xe0,4<x<5} 3) Which
1) 3) 1) The 1)I-N= N, the ELet {i.-i} 1) n(4nBnC) )4
l isn(4) If
a {x:xe and N 4= limbs.battle
college (x:a=5,
r:r'= set set
a ofthe ofthe be (r:x=1}
e B (4
23.nAnBoC)=6.n4ocoB)=9.nAocoB=3then the the = cach. that
The N,aN are of The 70%
of N,4<x<
following UBUC)AnB even set
number 300
5,xe following disjoint
minimum of Itm the
xe integers of the number
students, N} non and is
ax:x = Z} combatants an
5} is non BUC 2) 12
2) 2)2
integer,
2) of 12N 2) doesnot a 2)B NoN, =¢
negative - (-1.1}2) B
2)andP 2)S value of
newspaper
at singleton empty =(x:=} clements
most every e
N} the of then
have integers. 'xlost
20 then sets. nCac set is
student set? n in
is a then one
the proper of the isthe
prime eve, set
reads set A the I 4AB
subset - EoP 3) 15
3) 80% 3)3 S
exactly25
3) &NoGN
24N 3) {x:xe
4) 2)
Q! (x:x=6,xe
4) N;
2) 3) (A BoC 3)
cquals numbers.
=Then set 3) 3)6 is
these
none of - of
(-1.1.i.-i} = 105
newspapers (x:x B)= an
integers, car. and
i +3r+2 75% that
and N, an S
is
=0,xeN} the arm. split
every 4)7
48N 4) AUB 4) 4) =l-{0;
NAN, 4) set {1.i}4) 4)
||
none 4) 85% 4)4 into
none of
newspaper
non a subsets. n
of leg.
these positive - x%
is lost ImI
read integers. valueof all
by the 2
60
TICS 67) 66) 65) 64) 63) 62) 61) 60) 59) 58) S7) 56) $5)
)x-ye A Let
1) 1)
3
n(A)
3 = If Ifn(A)25
1) Ac
1)
BP(A) Ifn(A) AnB
{3,4,10}1) IfAUBe 1) Let 2"-1 1)
IfAis 1)3 Set physics.
4, n(A) 1)IfA AUB? 22
1) In I)A If 1)A=BI1A,
A A A A aclass
be ,
=25,n(B)=15. = and and and A=0.|0
a 4, P(B) =3 {2, = finite a B,
set .., B Bare Then
and 3, B 100 C
represented 4. be have be
A00 > two set two students, three
n(B) 8, the
10} containing 9 number then
are sets disjoint and
xyE 2)
A 2)5
=6 sets
10
2)
then BeA 2) .B=(3, 2) 6 power
by sets 2)9and AnBcAU 2) B then 2"-12) 2)6 33
2) 55
(2, n(4 2) elements (0.{0}. 2) A B=C 2)such
the such maximum n sets, studentsof
AcB.then 8, clements, students set that
square 5.4. Uthen
that 10} B) respectively, of
n(A) have Ais 4
of n( bnumber 10, who )B=
natural 4) then
number 12} + passed
number n(B) have AUC
x+re1 3) 3)4 =1+2 3) 3)6 ,C= 3)
15of 3) 3) -2 3)
2" n(A B)
3) 3)1 What passed
number AnB = 10
3) in 3) 3)
and elements A of (4. (4.5. Mathematics
=
B elements of can Aand
5. proper 0.(o}.{10}}.
in =C
and 4 6} 6, = be physics 4B=AoC
c in AUB the
x A, 12, subset
.y in minimum
C....cA,p0 AwR0S 14}
are 4UB only and
4)6 4).40 none 4) then ofAis 67
A}
4)EA any none 4) 4) none 4) 2"
4) n(B) 4) is
0sequal 4) 9 45
4)
students 4) nen
4)
two {3, (An number
5, none A
=
elements and B
to 14}
B)u(AnC)= have of =C
ofelements these
4
Passed
ofA =A
ther ther in
 4) 3) 2) SOLUTIONS
1)HINTS
AND 70) 69) 68)
TICS
I-PUC

Ans : Let Ans:(3) .Ans:(2) n(BUC)=5 1)

Since, Number x'+4x*Ans:(3) B=
A-
1) IfAUBUC 1) ItA=r:NEZ,-2Sx
Let
D==1.xe then R
100 15x25x the =(x):x+y=4, 4 U
nn4=n(4,) /=l0
10-3=7 = (2) Number be
A,X=30000 100 5x 100 105x +
total R-{0,-3,-1}A=
.:. +3x= the
of
cA,cA,c...c =1500 population elements universal
-1500 + of
X=1500 elements x(x
in + n(D)-6
2)
in S=10 2) ABOC 2) set
100 65x the in 4x+3) and B-
A=10×9=90
= town
2}.B
1
x,yeZ}
AUBUC=U.Then(4-B)U(B-)u(C-1)| =
A, and = A
=
then is x(x+3)(x+ B {x:e =
x 4
|=10 70 ={(x, and
|
|4 B Z.0%xS
):x,yeS.x*y} 1) =(.y):*+y'
AnB) 3) AU(BoC) 3) 3)
n(
=
Ao
4(BUO)=5
3}.C=

=9, x:Ne

AB= xyeZ}
4)
A 4) N,1<r<
sectoi An(BUC)
4)
then none
2:
of
these
 13) 12) 11) 10) 9) 8) 7) 6)
Ans: (2) LCM Ans: () Ans: Ans:(2)
n(AnB') From Ans:(3) and From
data :(3)
Ans
SoLet Ans(3) : :(2)
Ans
Given total total n(A) nowA'UB'= n(S)+ Ans(2) :
of{3. (3) data,
number
number A'OaUB)B']= n(P)=
7} = data m,=
300 = e'
m=6.n=3 =-1 N,
=21 n(AUB)= =x 2" n(B) ANB= 4UA'=
700- 2"-2"= of of 33+8
=subsets
Which 56+2"subsets =n
[200
=n(U)- ’-=1 N 4|
+ is 56= ofA
not of
300 AU[An
possible 64-8=2° isB is =NoN=
N
=AUAB)
o(4UB')
=(4'U A)
n(AUB) - 2" yn
100]
-2 B)U(Bn
reR x=-1
)BUA' =
AUB) which B')
is
not BB'=) (:
possible
 19) 18) 17) 16) 15) 14)

Ans: () Ans:(4) A=
Ans:(2) 100 Ans: ie.,
T99 N39 n(0UA) X=((1.2. Ans:(4)
Y={(2.3.5). so But Total
{0,tl,t2. Ans:(3)
f3.#4}A=
nU4 n( XoY=(14.5).(2.3.5)} remove
R-[0.2) np{p{P{PMD}}}
=2'=nP{P(M}=2'
nP{P{PA}}}-2'
16 =4 -2P(A) 39<xs63 > 512 atleast
number
63+
(4) -
and > 10 two
=2" 76X 7).(1.3.6).
[4,]= clearly n(0) = . of
-1<x<l =1 (1. 502clements,
{1} subsets
rS0x2
or t
n(4)n(OnA) 4,
x<63 5)
(2}.(3}.-}.-2}
99 ..
(5(1.4. =
2°
-| =
Or 1,
4)}
5).(2. 512
10 = I-1>1
3.
5)}
.(-3}.

(-4}.(4}.0;

-1
0 1
1
2
20) Ans: (4)
AAB =(4- B)U(B- A)=}5.6}-I,5.6}
21) Ans: (4)
DIAGRAM

22) Ans: (2)

A=(S.T. U, D. E. N
B=(P,R.0. G, E. S
C= {C.O. N, G R.U.E, T}
BAC= (0,R,G, E}
AU(BAC)={S.T,U,D,E,N,0.R.G}
23) Ans: (2)
x+1=0 ’ =-1 x=tieR

24) Ans: (3)

n(BUC) = n(B) + n(C)-nBOC)
= 50+20 - 10 = 60

25) Ans: (4)

n(AB)'= n(U) -n(AU B)
- 20-[12 +9 - 4]
= 20 - 17
=3

26) Ans: (1)

{1.4.9,16,25.... =1,2,3,4,5,..
(x:*=n,ne N}
27) Ans: (2)
n(S)+ n(P) = 32+8= 41

28) Ans: (2)

2 =8x 24
2
-=8
24
2-4 =8-2
p-q=3
29) Ans: ()
(2. 3, 5, 7}
n(p(A))= 2' = 16

30) Ans: (2)

5+x>10

(6.7.8,9}
31) Ans: (4)
not welldefined

32) Ans: (3)

33) Ans: (3)

A={M , A,T. H,E.I.C.S}.B= (S,T, A,.C?
A- B={M .H,E} .B-A=!
ANB =(4- B)(B- A) =(M .H.E}U!
={M,H ,E}= A- B
34) Ans: (4)
n(C) =224 . nH) =240, n(B) =336, n(Bo H)=64, n(Cn B) =80 ,n(CAH)= 40.
n(CoHnB)= 24. n(U) = 800
n(BUCUH) = n(B) +t n(C)+ n(H)- n(BoC)- n(CoH) - n(Bo H) +n(BoCoH)
=336+ 224 +240 80 40 -64+24
= 640
n(BUCUH)' = n(U) - n(BUCUH)= 800 640 = 160

35) Ans: (1)

n(AUB) = 36, n(AnB)=16, n(4- B) =15
n(A) =n(An B) +n(A- B) = 16+15 = 31
n(B) =n(AUB)+n(An B)- n(A)= 36+ 16-3|=21
36) Ans: (2)

37) Ans: (3)

38) Ans: (3)

set is subset of everyset

I-PUC MATHEMATICS-,.
 39)
Ans:(4)
The number of clements in S bc k
Then n(A) +nA,)
+..+nA)=10k ------(|)
n(B, )+ n(B, )+ .... + n(B,) = 9k ---.2)
From(l) 30 ×5=1Ok > k=15
From(2) nx3= 9k ’ 3n =9x|5 > In=45

40) Ans: (2)

n(C) =25,n(T) = 20, n(Co)=10,n(U) =60
nAUB)'= nU)-nAUB) = n(U) -nA) +n(B) -nAn B)
= 60 -[25 + 20 10] = 25
41) Ans: (1)
An(Bo)= LCM of 12,3.5, =30N = 30.60.90...
42) Ans: (2)
2"-2" =112 ’ 128-16=112 ’ -2 =|12
m=7,n=4

43) Ans: (3)

Let n(A) = n,n( B)= n

n[Pa]=2".n[PB)]=2"
nP]=n[p)]+15
2" =2" +15
2"-2" =15
16-1=15 2-2° =15
m=4,n =0
44) Ans: (3)
Number ofelements in eachset = 1lm +2
Number ofelements in nsubsets n(1lm +2)
.. n(llm +2) = 105 = 21x5=3x7x5
n=3 ’ |lm +2=35 = IIm =33 > m=3

45) Ans: (1)

X=100-[30+ 20 +25 + 15]= 100 90 = 10
46) Ans: (2)

nAnBnC) =m(4)-|n An Boc )+{4nBoc)+nAn Bn)

= 23 -(6+9+3]=5
|-PUCMATHEMATICS
47)
Ans: ()
4= 11,-1}.B= 1. I..-8 1- B=.B-A=.
AAB =(4- B}U(B - A) =i.

48) Ans: (4)

NAN, =(N - N,)U(N, - N)= 1.2.3,.., ut.-2.-1I} =/-0;

49) Ans: (3)

(AUBUO)n4B nCnce=(4UBUC)OHCUBUCoce

=(BnC )u(Coc)=(Bo)ug= Bnc

50) Ans: (3)
A-(4- B)= A B=¢ |:. Aand B are disjoint

51) Ans: (2)

x=6 X=6 because 6e N

52) Ans: (1)

(x:xe N ,4<r< 5} is an empty set and it has no proper subset

53) Ans: (3)

8N O6N = LCM of 8.6 = 24N

54) Ans: (3)

Ifx bethe number of newS papers

then 60= 300 X= 25

55) Ans: (2)

56) Ans: (3)

57) Ans: (4)

nM) =55, n(P)= 67, n(MUP)= 100
n(MoP) =n(M) +n(P) -n(MU P)
- 55 + 67-100 = 22
n(P only) = n(P) - n(M P)=67- 22 = 45
I-PUCMATHEMATICS
 58) Ans: (4)
Minimunm number of AUB= max (n(A), n(B)}
=max (9,6} =9

59) Ans: (2)

Aand Bare disjoint AoB= n(AnB) =0
n(AUB) = n(A) + n(B)

60) Ans:(1)

61) Ans: (2)

62) Ans: (1)

AB=(3,4,10},AnC= (4}
(AnB)u(4nc)=3,4,10}
63) Ans: (3)
Since AcB then n(AUB) =n(B) =6

64) Ans: (3)

65) Ans: (4)

Maximum number ofelements in A UB=nA) +n(B)= 25 +15 = 40
66) Ans: (2)
|00

since 4, A, C.. A,00 ’ 4 = A, = A

|=3

Given n(A) =i+2

n(A) = n(A,) =3+2 =5

67) Ans: (2)

A={x:x isasquare ofa natural number
A={1,4,9, 16, 25, 36, ...
Let 1,4 E Athen
1|4=-3¢ A
1+4=5 g A

I4=4 EA
xyE A Vr,ve A

|-PUCMATHEMATICS
68) Ans: (4)

A=-2,-1,0,1. 2.3; . B= 0.1,2.3} .C=1.2}.D=1.-;

Now BUC=0.1.2.3} n(BUC)=4+ 5
nD) =2 +5

AUBU0=2,-1,0.1.2,3; nUBUO)65

(69) Ans: (2)

(4-B)U(B- A)U(C-1) isthe shaded region
f4- B)u(B-O)U(C-4) is the unshaded region i.e.. ABOC

70) Ans: (2)

A=():*+=4,x.yeZ}=0.2).(2.0)} C
B=(*.r):x +y =9. x.ye Z} = (0,3).(3,0);
A- B=A
AnB=
B- A= B