SETS
In mathematics set is defined as the collection
of well defined object which can be separated
distinctly.
For instance,
S = {2, 4, 6, 8} is collection of the even integers.
A set can be explained in different ways:
i. Listing method: A = {a, b, c, .z}
ii. Descriptive method: N = {the natural
numbersfrom 1 to 50}
iii. Set builder method: A – B= A -(A∩ B )
iv. Venn – diagram
Universal sets
A universal set is the collection of all objects in
a particular context or theory. All other sets in
that framework constitute subsets of the
universal set, which is denoted as letter U. The
objects themselves are known as elements or
members of U.
Subsets
The set made by elements of the universal sets
is called subsets of the universal sets
For example
U = {1, 2, 3, 4, …………..50}
A = {even integers from 1 to 50}
B= {odd numbers from 1 to 50}
Here, A and B are the subsets of U
Overlapping sets
Two sets are said to be overlapped if they have
same element in common.
A∩ B = {6}
A and B are overlapping sets.
Disjoint sets
Two sets are said to be disjoint sets if there is
no element in common.
Cardinality of the sets
The number of the elements in the given sets is
known as cardinality of sets.
A = {1, 2, 5,}
B = {5, 3, 4}
AUB = {1, 2, 3, 4, 5}
n(A) =3
n(B) = 3
n(AUB) = 5
Cardinality of the three sets
LetA and B and C represent three sets
n(AUBUC) =n(A) + n(B) + n(C) - n(A∩ B) -- n(B∩
C) - n(C∩ A) +n(A∩ B∩ C)
Operation of sets
Union of sets
The set which includes elements of A and B is
called union of the sets.
A = {1, 2, 5,}
B = {5, 3, 4}
AUB = {1, 2, 3, 4, 5}
Or, AUB= {x: xϵ A or xϵ B}
n(A) =3
n(B) = 3
n(AUB) = n(A) + n(B) - n(A∩ B)
= 3 +3 -1 = 5
Intersections of sets
If the elements of set belongs to both Sets A
and B, it is called intersection of A and B.
A = {1, 2, 5}
B = {5, 3, 4}
A∩ B = {5}
n(A∩ B) = n(A) – n (A) = 3 -2 = 1
Or, A∩ B = {x: xϵ A and xϵ B}
Complement of sets
The set that contains all the elements of
universal sets except the given set A is called
complement of the set A . It is denoted by A̅
Difference of sets
If A and B are the two sets , the difference of
the dsets is the elements of the set thst
includes only in one set .
A – B= A -(A∩ B )
B - A= B-(A∩ B )
Example
Examples1
In a group of 200 students who like game, 120
like cricket game an 105 like football game. By
drawing Venn diagram find
i. how many students like both the games ?
ii. How many students like only cricket?
Soln
n(U) = 200
C and F denote the students who study Cricket
and football respectively.
n(C) =120
n(F) = 105
n( C ∩ F)=?
We have
n(CUF)=n(C) + n(F) -n(C∩ F)
200=120 + 105 -n(C∩ F)
n(C∩ F)= 25
n (C) = n(C)-n( C ∩ F)= 120 – 25 = 95
Examples 2
In the certain examination, 50% students passed
in account, 30% passed in English, 30% failed in
both and 25 student passes in both subjects. By
drawing Venn – diagram, find the number of the
students who passes in account only.
soln: Let total number be x A and E denotes the
students who study account and English
respectively
n(A) =50%
n(E) = 30%
n( A U E) = 30%
n( A U E) = 25
We have
n(U)= n(A)+ n(E)+ - n( A ∩E)
100%= 50 % + 30% + 30 % - n( A ∩E)
n( A ∩E) = 10%
According to the question,
10% of x = 25
x= 250
The number of the students who passed in
accounts only = 40% of the 250 = 100
Examples 3
In a survey it was found that 8.%people like
oranges ,85% like mangoes and 75% like both
But 45 people like none of Them. Drawing Venn –diagram , find the number of the people
who were in the survey.
Soln:
Let O and M be the number of people who like
oranges and mangoes respectively.
n (U) =n(OUM) +
100%=n (O) + n (M) - n(O∩M) +
100% =80% + 85% - 75% +
= 10%
According to the question,
10%of total number(say x)= 45
Or, x= 450
Question for Practise
Important Questions for SEE
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1. If n(A) = 40, n(B) = 50 & n(A∩B) = 15, find the
value of n(A∪B)
2. n(U) = 80, n(P) = 50, n(Q) = 30 & n(P∩Q) = 10.
Draw the Venn diagram to illustrate the above
information and find the value of n(P∪Q).
3. If n(A) = 45, n(B) = 65 & n(A∪B) = 85 then
a. Find the value of n(A∩B)
b. Find the value of n₀(B)
c. Show it in venn-diagram.
4. A survey conducted shows that 75% like milk
and 60% like curd. If they like atleast one of
them
then]
a. Show it in a Venn diagram.
b. What percent were there, who like both?
c. What was the percent, who like curd only?
5. In survey of 170 girls it was found that 18
girls can neither dance nor can sing, 96 can
dance and 69 can sing. By representing these information in a Venn diagram, find the number of girls who can both dance and sing.
6. Out of 100 students, 80 passed in science, 71
in Mathematics, 10 failed in both subjects and 7
didn't appear in an examination. Find the number of students who passed in both subject using venn diagram.
7. In a survey of 120 students, it was found that
17 drink neither tea nor coffee. 88 drink tea and
26 drink coffee. By drawing Venn-diagram, find out the number of students who drink both tea and coffee.
8. In a survey of 2400 tourists visited in Nepal, it
was found that 1650 liked to visit Bhaktapur, 850
liked to visit Lalitpur and 150 did not like to visit both places:
a. Represent the above information in a Venn-
diagram.
b. How many were there who liked to visit both
places?
c. How many were there who liked to visit
Bhaktapur only?
9. In a survey of 2000 tourists visited in Nepal, it
was found that 1125 liked to visit Pokhara, 750
liked to visit Janakpur and 250 did not like to visit both places.
a. Represent the above information in a venn-
diagram
b. How many were there who like to visit both
places?
c. How many were there who liked to visit
Janakpur only ?
10. In a school of 80 students of class X were
asked what they would like milk or tea 60 said
they would like tea, 50 said they would like milk and 10 said they would like neither milk nor tea. By drawing Venn-diagram, find the number of students.
a. (Who like both tea and milk?)
b. (Who like milk only?)
c. (Who like tea only?)
11. In a survey of group 100 students 55 like to
read Muna magazine, 45 like Yuba Manch and
14 do not like to read either of them, Draw a Venn diagram and give of answer to the following How many of them like to read
a. Muna magazine only
b. Yuba Manch only.
c. Both of them.
12. In a survey of 60 students, 30 drink milk, 25
drink curd and 10 drink milk as well as curd
then
a. Draw a venn -diagram of above information.
b. Find the number of students who drink
neither of them.
13.Out of 100 students in a S.L.C. Examination,
80 passed in math, 60 passed in science and 50
passed in both subjects.
a. Draw a Venn diagram to illustrate the above
information
b. Find the number of students failed in both
subjects.
14. In a class of 25 students, 17 like Volleyball,
15 like Basketball and 10 like both games, Illustrate it with a Venn-diagram and find the number of students who don't like any of the games.
15. In a survey of 200 students 125 liked to
admit in science faculty, 100 in humanity faculty
and 40 like to admit either of faculties and the rest were found not to be admitted in both faculties.
a. Show the above information in a venn
diagram.
b. Find how many were there who don't like to
admit in both faculties.
16. Out of 5000 Japanese who are on Nepal tour,
40% have already toured India 30% have also
toured Pakistan. If 10% of them have toured both the countries by drawing Venn-diagram find the
number of tourists those who have not yet visited both the countries.
17. Out of 1350 candidate 600 passed in Health,
700 in social, 350 in Nepali and 50 did not
passed in all three subjects. If 200 passed in Health and social, 150 in Health and Nepali, 100 in social and Nepali.
a. How many candidates passed in all subjects?
b. Illustrate the above information in a Venn-
diagram.
18.G, S & K represent the people who read
Gorkhapatra, Samacharpatra & the Kantipur
daily. If
n(G∪S∪K) = 100%, n(G) = 50%, n(S) = 55% n(K)
= 35%, n(G∩S) = 15, n(S∩K) = 10%, n(G∩K) =
20%, then what percentage of the people read
all these three daily papers?
19. Of the total candidates in an examination,
40% students passed in math 45% in science and 55% in Health. If 10% passed in math and science, 20% in science & Health and 15% in Health and Maths.
a. Draw a Venn-diagram to show the above
information
b. Calculate the percentage. Who passed in all
three subjects?
20. If n(A) = 65, n(B) = 50, n(C) = (A∩B) = 25, n
(B∩C) = 20, n(C∩A) = 15, n(A∩B∩C) = 5 and n(U) = 100, find the value of n( A∪ B ∪C )
21. In a survey of 100 people 65 read Kantipur,
45 read Gorkhapatra, 40 read Himalayan times,
25 read Kantipur as well as Gorkhapatra, 20 read
Kantipur as well as Himalayan times, 15 read
Gorkhapatra as well as Himalayan times and 5 read all three news papers.
a. Draw the Venn-diagram to illustrate the above
information.
b. How many people don't read all three news
papers.
22. In a survey of a group of students it was
found that 60 like to listen poems, 45 liked to
listen stories, 35 liked to listen both and 10 did not like to listen both. Then
a. Present the above information in a Venn-
diagram.
b. How many students were surveyed?
c. How many students were liked to listen the
story only?
23. In a survey of a group showed that 60 liked
tea, 45 liked coffee, 30 liked milk, 25 liked coffee
as well as tea. 20 liked tea as well as milk, 15 liked Coffee as well as milk and 10 liked all three how many were asked this question. Solve by drawing venn-diagram.
24. Among the candidates appeared in an
examination, 80% passed in English, 85% in
Maths and 75% in both English and Maths. Find the total number of candidates if 45 candidates were failed in both English and Maths.
25. By filling the following information in a venn
diagram, Find the cardinal number of n (U). n(L)
= 14, n(M) = 13, n(N) = 22, n(L∩M∩N) = 6, n(L∩M) = 7, n(M∩N) = 9, n(L∩N) = 11,
n( L ∩ M ∩ N ) = 4
26.In a school 60 students passed in Mathematics, 45 passed in Science and 30 passed in both subjects.
a. How many students were there in the school?
b. How many students have passed in
Mathematics only?
c. How many students had passed in Science
only?
d. Represent the above information in a Venn-
diagram
27. In a college, 65 are studying in mathematics,
50 students are studying in science and 35 students are studying in both subjects.
a. How many students are studying in the
college?
b. How many students are studying in
Mathematics only?
c. How many students are studying in Science
only?
d. Represent the above information in a Venn -
diagram.
28. In an examination, it was found that 55%
failed in Maths and 45% failed English. If there
were 35% passed in both subjects.
a. What percent failed in Maths only?
b. What percent failed in English only?
c. Represent the above information in a venn
diagram.
29. In a survey of a village, it was found that
85% of the people like Dashain Festival and 60%
like Tihar Festival. If there were 5% people who did not like both festival:
a. What percent liked Dashain Festival only?
b. What percent liked Tihar Festival only?
c. Represent the above information in a Venn-
diagram.
30. In a school, all students play either volleyball
or football or both. 300 play football. 250
volleyball and 110 play both game draw a Venn diagram to find.
a. Number of students who play football only.
b. Number of students who play volleyball only.
c. The total number of students in the school.
31.In a survey of a community of 240 people, it
was found that 151 people could speak Newari
language, 95 could speak English language and
14 could speak neither of the languages.
Represent the above information in a Venn-
diagram supposing x as the number of people
who can
speak both languages.
a. Find the value of x.
b. Find the number of people who can speak
English only.
32. 40% of the students of a school play football,
30% play volleyball and 20% play both. If 90 students play neither football nor volleyball use Venn- diagram and find the number of students in the school.
Also find the number of students who play only
football.
33.Out of 20 staffs in an office, 15 can speak
local Newari language other than Nepali
language, 8 can speak Bhojpuri, while 3 can not speak any local languages. By drawing venn diagram, find the number of staffs.
a. Who can speak at least any one local
language?
b. Who can speak any one local language only
out at the two.
34. 40 students were asked what they would like
milk or curd. 30 said they would like curd, 25
said they would like milk and 5 said they would neither like milk nor curd. By drawing Venn diagram, find the number of students.
a. Who like milk only?
b. Who like curd only?
c. Who like either milk or curd, any one only?
35. Set P & Q are two subsets of a universal set
U. Illustrate the given information in a venn -
diagram & solve the following problems.If n(U) = 40, n(P) = 25, n(Q) = 18, n(P∩Q) = 5 n(P∪Q) = ?, n(P - Q) = ? n(Q-P) = ?
36. If U = {X:X is a positive integer less than 16}
A = {Y:Y is a prime number} B = {Z:Z is an odd
number} then write down the cardinal value by
drawing venn diagram.
a. n(A∩B)
b. n( A∪ B )
c. n(A - B)
d. n[U-(A∩B)]
37. By asking question in a group, 110 answered
TB spread through drugs, 75 said through
smoking and 60 said through food. Among them, 25 said from both drugs and smoking, 10 said from smoking & food & 10 said from drug and food, while 5 said from all three.
a. How many pedestrians were involved? Solve
by drawing Venn-diagram.
b. How many said TB spreads through drugs
only?
38. In a group 30 students read Math, 24 read
Economic, 22 read Statistics 14 read Math only,8
read Economics only, 6 read Math and Statistics only, 2 read Math and Economics only and 8 read neither of them
a. Find the number of students in their group?
b. How many students read Economics and
Statistics only?
c. How many students read all three subjects?
d. Show all the information in Venn-diagram.
39. In a class of 25 students, 12 have taken
Mathematics, 8 have taken Mathematics but not
Biology. Find the number of students who have taken Mathematics and Biology and those who have taken Biology but not Mathematics. Solve by making a Venn-diagram.
40. In a interview of 50 people. 15 liked milk but
not coffee, 5 liked coffee and milk and 5 did not
like both. How many liked coffee only? Represent all result in venn diagram also.
41. Each of a group of 20 students study at least
one of the three subjects Nepali, English and
Maths. All those who study English also study Nepali 3 study all three subjects. 4 students only study Nepali. 8 students study English. 14 students study Nepali. Draw Venn-diagram to illustrate above information.
a. How many students study only Nepali and
Maths but not English?
b. How many students study only Maths?
c. Show all the information in Venn diagram?
42. Each group of 25 students studies at least
one of the three subjects Nepali, English and
Maths. All those who study English also study Nepali, 4 students study all three subjects, 5 students study only Nepali. 7 students study English and 15 students study Nepali.
a. Draw the venn-diagram to illustrate above
information.
b. How many students study Nepali and Math
but not a English?
c. How many students study math only?
43. In a group of students 18 read Account, 19
read Maths, 16 read Science. 6 read Account
only, 9 read Maths only, 5 read Maths and Account only, 2 read Maths and Science only. By using venndiagram, find the following
a. How many read all subjects?
b. How many read science only?
c. How many read account and science?
d. How many students are there altogether?
44. Out of 50 students in a class like Mathematics or Science or both. Among them 20 like both subjects and the ratio of Maths and Science is 3:2
a. Find the number of students who like Maths
only.
b. Find the number of students who like Science
only.
c. Illustrate the above information in a venn
diagram.
45. In a class of 75 students, 30 students liked
Nepali but not Science and 25 students liked
Science but not Neplai. If 10 students did not like both, how many students liked both subjects and
represent the above information in venn-diagram.
46. In a class of 55 students, 15 students liked maths but not English and 18 students liked English but not Math. If 5 students did not like both, how many students liked both subjects and represent the above information in Venn-diagram.
47. In an election 1400 voters did not cast their
votes and 400 votes are declared invalid. Two
candidates A and B in the election A defected B
by 60 votes. It was found that A secured 2400 of
total rolled votes. Find the number of voters.
48. In a survey among 100 people, 50 liked
coffee, 30 liked milk, 40 liked tea, 20 like coffee
only, 25 liked tea only, 10 liked tea and coffee and 5 liked tea, coffee and milk all. Using venn-
diagram, find the number of people who liked neither of these.
49. An ice-cream shop sold three type of ice
creams A, B & C out of the customers questioned, 470 liked A, 300 liked B, 350 liked C, 130 liked A and B, 120 liked A and C, 70 liked B and C, 50 liked all three.
a. Draw the venn-diagram to illustrate above
information
b. How many liked exactly two types?
50. A survey was done about mobile and
telephone facility 200 people had telephone
facility, 25 had both facility 135 people did not have telephone facility and 300 did not have mobile facility?
a. How many participants in the survey?
b. How many had mobile facility only?
51. In a class of 65 students, 10 students liked
Maths but not English and 20 students liked
English but not Maths. If 5 students did not like both how many students liked both subjects? Solve by Venn-diagram.
52. In a result of first terminal examination of
class ten 20% students are passed in all three
subjects 46% failed in Maths, 57% failed in English, 42% failed in dance, 35% failed in Maths and English, 25% failed in English and dance, 15% failed in dance and Maths.
a. Draw venn diagram to show above
information
b. Find how many percent failed in all three
subjects
53. In a village of 140 house, 60 believe in
Buddhism, 70 in Hindu Religion and 45 in other
religion. Among then 17 house don't find any difference in Hindu and Buddhism, 18 house don't find any different in Hindu and other religion while 16 houses don't find any difference between Buddhism and other religion. If 6 houses don't believe in any religion, find how many houses do not find any difference in any religion. Show it in venn-diagram.
54. In a group of players 40 play Volleyball out
of which 25 play volleyball only. 10 play football
only and 8 play badminton only 27 play football, 30 play badminton out of which 15 play volleyball and badminton both. Using venn-diagram find.
a. How many play all three games?
b. How many play two games only?
55. In an interview of 60 people space times,
Rajdhani and Himalayan times are liked by 45,
30 and 15 people respectively. If the total number of people who liked only two magazines in 22 and they liked at least one of the magazines. Find the number of people who liked all the magazines.
56. In an examination 48% failed in science, 39%
in account and 33% in History, 12% in science
and Account, 9% in science and History and 13% in Account and History and 3% in all three
subjects. Draw Venn-diagram to represent the given information and find the.
a. Percentage passed in all three subjects.
b. Percentage failed exactly in two subjects.
c. Percentage failed exactly in one subject.
Answers
1. 75
2. 70
3. 25, 40
4. 35%, 25%
5. 13
6. 68
7. 11
8. 250, 1400
9. 125, 625
10. 40, 10, 20
11. 41, 31, 14
12. 15
13. 10
14. 3
15. 15
16. 2000
17. 100
18. 5%
19. 5%
20. 15
21. 5
22. 80, 10
23. 85
24. 450
25. 32
26. 75, 30, 15
27. 80, 30, 15
28. 20%, 10%
29. 35%, 10%
30. 190, 140, 440
31. 20, 75
32. 180, 36
33. 17, 11
34. 5, 10, 15
35. 38, 2, 20, 13
36. 5, 6, 1, 10
37. 205, 80
38. 54, 6, 8
39. 4, 13
40. 25
41. 2, 6
42. 3,10
43. 3, 7, 7, 36
44. 22, 8
45. 10
46. 17
47. 6540
48. 20
49. 170
50. 335, 10
51. 30
52. 10%
53. 10
54. 10, 12
55. 4
56. 11%, 25%, 61%