Whole explanation about set theory

1. Set Theory Intro
• Definition
• Universal Set and Null Set
• Subset and Superset
• Powerset, Proper Subset,
Improper Subset
• Complement of a Set
• Cardinal Number
• Union and Intersection
(also disjoint sets)
• Difference of Sets

3. •70% of the people like Coffee, 80% of the people
like Tea; then at least what % of people like
both?
Ans - 50

4. •70% of the people like Coffee, 80% of the people
like Tea, 85% of the people like Milk; then at
least what % of people like all three?
Ans - 35

5. •The set A of odd positive integers less than 10
can be expressed which of these options?
• (a) {1, 3, 5, 5, 7, 9}
• (b) {5 Elements}
• (c) {1, 3, 5, 7, 9}
• (d) None of these
Ans – (c)

6. •The members of the set S given
S= {x | x is the square of an integer and x < 100}
is:
Ans – S = {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}

7. •The roster form of the following set E = The set
of all letters in the word TRIGNOMETRY is
Ans – E={T,R,I,G,N,O,M,E,Y}

8. •Let A = {0, 1, 3, 5},
B = {5, 6, 1, 3, 9} and
C = {0, 1, 2, 3, 9, 13}.
Then, (A n B) u C =
Ans - {0, 1, 2, 3, 5, 9, 13}

9. •Convert the Set A, given A = {3,-3} into set-
builder form
A= {x:x is an integer and x2 - 9=0}

10. • Study the given figure and
answer the questions given.
1. How many doctors are
neither artist nor players?
2. How many doctors are
both players and artists?
3. How many artists are
players?
4. How many players are
neither artist nor doctors?
1.17 | 2. 3 | 3. 25 | 4. 25

11. Given here are five diagrams one
of which describes the
relationship among the three
classes given in each of the five
questions that follow.
• Statement A: Elephants, tigers,
animals
• Statement B: Administrators,
Doctors, Authors
• Statement C: Platinum, Copper,
Gold
• Statement D: Gold, Platinum,
Ornaments
• Statement E: Lawyers, Parents,
Gold
b | e | d | e | c

12. •A set has n elements, then the total number of
subsets are
•A set has n elements, then the total number of
proper subsets are
2n | 2n - 1

13. •The sets A & B have 6 & 9 elements respectively,
such that A is proper subset of B, then the total
number of elements in (A n B) are
•The sets A & B have 5 & 9 elements respectively,
such that A is proper subset B, then the total
number of elements in (A u B) are
6 | 9

14. •In a group of 60 people, 27 like cold drinks and
42 like hot drinks and each person likes at least
one of the two drinks. How many like both Cold
drinks and Hot drinks?
•If A & B are two sets such that n(A)= 15, n(B)=
21, & n(AuB) = 36 then n(AnB) equal to
9 | 0

15. In the Mindwork club all the members participate either in the
Tambola or the Fete. 320 participate in the Fete, 350
participate in the Tambola and 220 participate in both. How
many members does the club have?
At the birthday party of Sherry, a baby boy, 40 persons chose
to kiss him and 25 chose to shake hands with him. 10 persons
chose to both kiss him and shake hands with him. How many
persons turned out at the party?
450 | 55

16. •In a Group of 300 people, 150 can speak French &
200 can speak German. How many can speak both
French & German.
•In a Group of 300 people, 150 can speak French &
200 can speak German. How many can speak both
French & German, given each person in the group
speaks alteast one of the two languages?
CBD | 50

17. There were 100 students in the library
who responded to how they
completed their research paper.
• 18 students only used the periodicals
• 29 students used the web and books
• 15 students used books, the web,
and periodicals
• 40 students used books and
periodicals
• 20 used the web and periodicals
• 60 students used books
• 7 students did not use the web, nor
books, nor the periodicals.
How many used only the web?
10

18. Concerning the first 41 presidents
of the United States we know the
following facts: Eight held cabinet
posts, 14 served as vice-president,
15 served in the U.S. Senate, 2
served in cabinet posts and as vice-
president, 4 served in cabinet posts
and in the U.S. Senate, 6 served in
the U.S. Senate and as vice-
president, and 1 served in all three
positions. How many presidents
served in:
• None of these 3 positions?
• Only in the U.S. Senate
• At least one of the three position?
• Exactly two positions?
15 | 6 | 26 | 9

19. How many numbers are there between 1 and 100 that are not divisible by 2, 3 and
5?
In a survey of 130 people, the following data were collected: 106 people subscribed
to the newspaper, 29 people subscribed to magazines, and 17 people were
members of a mail CD club. Seventeen subscribed to both the newspaper and the
magazines, 5 people subscribed to magazines and were members of a CD club, and
10 people subscribed to the newspaper and were members of a mail CD club.
Three people subscribed to both the newspaper and magazines and were members
of a mail CD club. How many people did not subscribe to any of the three?
25 | 7

20. A school has 63 students studying
Physics, Chemistry and Biology. 33
study Physics, 25 Chemistry and 26
Biology. 10 study Physics and
Chemistry, 9 study Biology and
Chemistry while 8 study both
Physics and Biology. Equal numbers
study all three subjects as those
who learn none of the three.
• How many study all the three
subjects?
• How many study only one of the
three subjects?
3 | 39

21. In a competition, a school awarded medals in different categories. 36 medals in
dance, 12 medals in dramatics and 18 medals in music. If these medals went to
a total of 45 persons and only 4 persons got medals in all the three categories,
how many received medals in exactly two of these categories?
In a class of 120 students numbered 1 to 120, all even numbered students opt
for Physics, those whose numbers are divisible by 5 opt for Chemistry and those
whose numbers are divisible by 7 opt for Math.
How many opt for none of the three subjects?
13 | 41

22. In a competition, a school awarded medals in different
categories. 36 medals in dance, 12 medals in dramatics and 18
medals in music. If these medals went to a total of 45 persons
and only 4 persons got medals in all the three categories, how
many received medals in exactly two of these categories?
13

23. In the CBSE Board Exams last year, 53% passed in
Biology, 61% passed in English, 60% in Social
Studies, 24% in Biology & English, 35% in English
& Social Studies, 27% in Biology and Social
Studies and 5% in none.
• Percentage of passes in all subjects is
• If the number of students in the class is 200,
how many passed in only one subject
• If the number of students in the class is 300,
what will be the % change in the number of
passes in only two subjects, if the original
number of students is 200?
• What is the ratio of percentage of passes in
Biology and Social Studies but not English in
relation to the percentage of passes in Social
Studies and English but not Biology?
7 | 46 | 50% | 5/7

24. In an office, were working in
at atleast one department is
mandatory, 78% of the
employees are in Operations,
69% are in Finance and 87%
are in HR. What is
• The maximum percentage
of employees working in all
3 departments?
• The minimum percentage of
employees working in all 3
departments?
67 | 34

25. In a school there are 200 students. 100
play Cricket, 50 play Hockey and 60
play Basketball. 30 students play both
Cricket and Hockey, 35 play Hockey
and Basketball and 45 play both
Basketball and Cricket.
• What is the maximum possible
number of students who play atleast
1 game?
• What is the maximum possible
number of students who play all the
3 games?
• What is the minimum possible
number of students who play atleast
1 game?
• What is the minimum possible
number of students who play all the
3 games?
130 | 30 | 120 | 20

26. There are 3 electives offered to the
students in class of 92 (the students
have a choice of not choosing any
electives).
60 students opted for Marketing, 80
for Finance and 50 for Systems, 40
opted for both Marketing and
Systems, 50 for both Marketing and
Systems and 45 for both Finance
and Systems.
What is the maximum possible
number of students opting for all 3
electives?
What is the minimum possible
number of students opting for all 3
electives?
37 | 35