class 10 sets

Mathematics
10

Sets Review

1.In SEE you will be asked only two sets
(previously it will be asked of three sets too)

2. Only problem solving question will be asked
from set so that students can easily solved it.

3. Before solving the questions please read the
question carefully so that no error occurs on
your answer

4. While solving the problem be sure to write
the given things.

Some Formulae used in sets

1.n(A⋃B) =n(A)+n(B)-n(A⋂B)

2.n(U)=n(A⋃B)+( ⋃ )
3. n( ⋃ ) =n(U)-n(A⋃B)

4. (A)=n(A)- n(A⋂B)
5. n(U)=n(A⋃B) [if ( ⋃ )=0]
ie. If none of them like…..

Practice time
1.If n(U)=85, n(A)=45, n(B)=55 &n(A⋃B)=65
then find (i)n(A⋂B) (ii) (A) (iii) (B)
(iv) n( ⋃ ) and show them in Venn diagram.
Solution
We have given ……..
(i)n(A⋂B)=n(A)+n(B)-n(A⋃B)

= 45+55-65

= 100-65

= 35

(ii) (A)=n(A)- n(A⋂B)
= 45-35
= 10

(iii) (B)=n(B)- n(A⋂B)
= 55-35
= 20

(iv) n( ⋃ ) = n(U)-n(A⋃B)
= 85-65
= 20

Now showing above information in
venn-diagram we get,

A BU

10 35 20

20

2. If n(U)=75, n(A)=40,n(B)=35 & n( ⋃ )=10
then find (i) n(A⋃B),(ii)n(A⋂B)(iii) (A) /n(A-B)
(iv) (B) . Show them in venn diagram
Solution
We have given …..
(i) n(A⋃B)=?

=75-10=65 [n(U)-n( ⋃ )]
(ii)n(A⋂B)=?

=40+35-65=10 [ n(A)+n(B)-n(A⋃B)]
(iii) (A) /n(A-B)=?

= 40-10=30 [n(A)-n(A⋂B) ]

(iii) (B) /n(B-A)=?
= 35-10=25 [ n(B)-n(A⋂B) ]

Now showing above data in venn-diagram.

A BU

30 10 25

10

3. If n(U)=48, n(A)=27,n(B)=18 and n(A⋃B)=36
then by drawing a venn diagram find the following
(i) n(A⋂B) (ii) n( ⋃ ) (iii) (A) and (iv) (B)

Solution

Above data can be represented in venn-diagram as

A BU

27-x x 18-x

12

From venn diagram
36=27-x+x+18-x
Or, 36=45-x
Or, x=45-36

=9
∴(i) n(A⋂B)=x=9

(ii) n( ⋃ )=?

= 12
(iii) (A)=?

= 27-x= 27-9=18
(iv) (B)= 9 [How?]

Students time

Solve previous question without using
venn-diagram and submit to your
teacher

THANK YOU

Day 5
Good Morning Students
Any Problems in previous Class?
Have you done your work?

Lets start our class.

Things to be remembered in sets
1.At least one /either n(A⋃B)
2.Both n(A⋂B)
3. Did not like any n( ⋃ ) /neither
4.Liked only one (A)+ (B)
5.Only (A)

Word Problems To Numerical Problems
4 Marks Sure

For Class 8/9/10

Out of 100 participants in a picnic, 70 drank milk
40 drank soup and 20 drank both.
(i)Represent the above information in venn-
diagram.
(ii)Find the number of participants who drink
neither of the drinks.
Solution
Let M= Participants who like milk

S= Participants who like soup
Then, n(U)= 100 n(M)=70 n(S)=40 and n(M⋂S)=20

n(M⋃ )=?

(i)Representing above information in venn-
diagram.

M SU

50 20 20

10

(ii) From venn-diagram
n(M⋃ )=10

Hence 10 participants drank neither of the drink.

In the survey of a community, 55% of the people
like to listen radio, 65% like to watch
television and 35% like to listen the radio as
well as to watch television:
(i)Show the above information in venn
diagram .
(ii) Find the percentage of the people who do not
like to listen the radio as well as to watch television

Solution
Let R= People who like to listen radio

T= People who like to watch television
Then , n(U)= 100% n(R)=55% n(T)=65%
n(R⋂T)=35% and n(M⋃ )=?
(i) Now showing in Venn-diagram we get

M SU

20% 35% 30%

15%

A survey of students of Gyanjyoti Higher Secondary school shows that 45 students like
mathematics and 41 students like science. If 12 students like both the subjects, how many
students like either mathematics or science?

Solution

Let, M = set of students who like mathematics S = set of students who like science

Then, by question, n(M) = 45 n(S) = 41 Number of students who like both math &
science = n(M⋂S) = 12

we have, n(M⋃S) = n(M) + n(S) –n(M⋂ S) = 45 + 41 – 12 = 74

In venn diagram

M SU

45-12 12
=33 41-12
=29

Good Morning Everyone
Day 6

Lets start our class

A survey carried among 850 villagers shows that 400 of them like to make a
water tank, 450 like to make an irrigation plant and 150 like to make both of
them. Represent the information in a Venn–diagram and find: i. the number
of people who like to make a water tank only. ii. the number of people who
like to make either a water tank or an irrigation plant. iii. the number of
villagers who like neither of them.

Solution We have given

Let, T = set of villagers who like to make a water tank I = set of villagers
who like to make an irrigation plant

Then, by question, n(U) = 850, n(T) = 400, n(I) = 450 n(T⋂I) = 150

(i) (T)= ? (ii) n(T⋃I)= ? (iii) n(T⋃ )= ? T I
In venn diagram

400-150 450-150
=250 150 =300

x

From Venn-diagram (i) (T)= 250 ,(ii) n(T⋃I)=250+150+300=700 and
(iii) n(T⋃ )= 850-700=150

Out of 20 teachers of a secondary school, 15 can
speak local Newari Language,8 can speak
Bhojpuri while 3 cannot speak any local languages.
By drawing a Venn-diagram find the no. of teachers:
(i) Who can speak both languages.
(ii) Who can speak at least one local language and
(iii)Who can speak only one local language.
Solution
Let A= set of teachers who can speak Newari

B= set of teachers who can speak Bhojpuri
Then n(U)=20 n(A)=15 n(B)=8 and n( ⋃ )=3

Now in Venn-diagram, BU
8-x
A

15-x x

3

(i) n(A⋂B)=?
From Venn-diagram
15-x+x+8-x+3=20
∴x=?
Hence 6 teachers can speak both languages.

(ii) n(A⋃B)=?
n(A⋃B)=15+8-6

= 17

Hence 17 teachers can speak at least one language.

(iii) The number of teachers speaking only one
language
= (A)+ (B)
= 15-x+8-x
= ?(put x=6)
Hence the number of teachers speaking only one
language are 11.

There are 45 students in a class.24 of them like
Cricket, 30 like football and 14 like both of them
(i) Illustrate this information in Venn-diagram.
(ii) How many students like both games?
(iii)How many students do not like both games?
(iv)How many of them like only cricket?
(v) How many of them like only football?
Solution
Let ,

C= set of students who like cricket
F= set of students who like football

(i) Representation on Venn-diagram

C FU

14

10 16

5

(ii) n(C⋃F)=?
=10+14+16 [From Venn-diagram]
= 40

∴ 40 students like both of the games.

(iii) n(C⋂F)=?

= 14
∴ 14 students do not like both games
(iv) (C)=?

= 10
∴ 10 students like only cricket.
(v) (F)=?

=16
Hence 16 students like only football.

In a class of 60 students,15 students like math only,
20 liked English only and 5 did not like any
Subjects, then
(i) Find the number of students who like both the

subjects.
(ii) Find the number of students who like at least

one subject.
Solution
Let,
M denote the number of students who like Math
E denote the set of students who like English

By question,
n(U)=60, (M)=15, (E)=20, n(M⋃ )= 5
Let n(M⋂E)=x

We know that,
n(U)=?

= (M)+ (E)+n(M⋂E)+n( ⋃ )
or, 60=15+20+x+5
or, 60=40+x
or x=60-40

= 20
∴ 20 students like both subjects.

(ii) The number of students who like at least one
subject =?

= 15+x+20 [How ? ]
= 35+20
= 55
Hence the number of students who like at least one
subject is 55.

In a class of 60 students,15 students like math only,
20 liked English only and 5 did not like any
Subjects, then
(i) Find the number of students who like both the

subjects.
(ii) Find the number of students who like at least

one subject.
Solution
Let,
M denote the number of students who like Math
E denote the set of students who like English

By question,
n(U)=60, (M)=15, (E)=20, n(M⋃ )= 5
Let n(M⋂E)=x

We know that,
n(U)=?

= (M)+ (E)+n(M⋂E)+n(M⋃ )
or, 60=15+20+x+5
or, 60=40+x
or x=60-40

= 20
∴ 20 students like both subjects.

(ii) The number of students who like at least one
subject =?

= 15+20+x [How ? ]
= 35+20
= 55
Hence the number of students who like at least one
subject is 55.

A travel agent says that 20% of his
customers have been to Australia, 50% have
been to UK and 35% have been to neither of
those countries. What percentage of the
customers have visited both the countries?
Solution
Let
A and B be the sets of customers who have
been to Australia and UK respectively.
Then according to question,

n(A)=20% n(B)=50% and n( ⋃ )=35%
n(A⋂B)=?
Let n(A⋂B)=x

We know that
n(U)=n(A)+n(B)-n(A⋂B)+n( ⋃ )

or, 100%=20%+50%-x+35%
..Calculate value of x yourself with process…
∴x=5%
Hence 5% of the customers have visited both
countries.

DAY-8

There are 45 students in a class.24 of them like
cricket, 30 like football and 14 like both of them.
(i) Illustrate this information in Venn-diagram.
(ii) How many students like either of the games?
(iii)How many students like both games?
(iv)How many students don’t like both games?
(v) How many of them like only cricket?
(vi)How many of them like only football?
Solution
Let ,

C= set of students who like cricket
F= set of students who like football

(i) Representation on Venn-diagram

C FU

14

10 16

5

(ii) n(C⋃F)=?
=10+14+16 [From Venn-diagram]
= 40

∴ 40 students like either of the games.

(iii) n(C⋂F)=?

= 14
∴ 14 students like both games
(iv) n(C⋃ )= ?

=5
∴ 5 students dont like both games.
(v) (C)=?

= 10
∴ 10 students like only cricket.
(vi) (F)=?

=16

Hence 16 students like only football.

In a college,65 students are studying mathematics,
50 students are studying science and 35 students
are studying both subjects. Find how many students
(i) are studying in the college?
(ii) Are studying mathematics only?
(iii)Are studying science only?
(iv)Represent all results in a Venn-diagram.
Solution
Let M and S denote the set of students who are
studying Mathematics and Science respectively.
Then according to question,

n(M)=65, n(S)=50 and n(M⋂S)= 35
(i) n(U)=n(M⋃S)=?

= 65+50-35
= 80
∴ 80 students are studying in the college.
(ii) (M)=?
= formulae?
= 65-35 [ n(M)-n(M⋂S)]
=30
∴ 30 students are studying mathematics only.

(iii) (S)=?
= 50-35
= 15

(iv) Representation on Venn-diagram we get

MS

30 35 15

DAY-9

In an election of a municipality, two candidates

P and Q stood for the post of Mayor and 40000
people were in the voter list . Voters were supposed
to caste the vote for a single candidate. 20000
people caste vote for P, 15000 people caste vote for
Q and 3000 people cast vote even for both.
(i)Show the above information in Venn-diagram.
(ii) How many people didn’t caste vote ? Find it.
(iii) How many votes were valid?
Solution
Let P and Q be the set of votes for candidates P and
Q respectively.

Then according to question
n(U)=40000 n(P)=20000, n(Q)=15000 and
n(P⋂Q)= 3000
(i)Representing above information in Venn-diagram.

PQ U

17000 3000 12000
8000

From Venn-diagram,
(ii) People didn’t cast vote=?

=n(P⋃ )
= 8000 [How?]
∴ 8000 people didnt cast vote.
(iii) No. of valid votes=?
No. of valid votes= (P)+ (Q)

= 17000+12000
= 29000
Hence 29000 votes were valid.

In a survey conducted among some people of
a group, it was found that 40% of them like
literature, 65% of them like music and 10%
of them liked none:

(i) Illustrate these information in a Venn-diagram
(ii) If there were 30 people who liked both of them,
find the number of people participated in the survey.
Solution
Let L and M denote the set of people who like
literature and music respectively.

Then by question,
n(L)=40%, n(M)=65%, n(L⋃ )=10%
Let n(L⋂M)=x%
(i) Now showing in Venn-diagram we get,

LM U

(40-x)% x% (65-x)%

10%

(ii) We know that n(U)=100%
or, 40-x+x+65-x+10=100
…………………………….
∴x=15
∴n(L⋂M)=15%
According to question,
15%=30
or, 1%= =2
∴100%=100*2=200
Hence 200 people participated in the survey.

Day 10