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# Sets

A set is a collection or a group of well – defined objects or the items having some common characteristics.
Well defined means – Meaning should be clear whether a particular object belong to the group or not.

Set Operations and use of venn diagram
There are mainly four set operations:

1. Union of sets
The union of set A and B is the set of all members that belongs to either A or B or both.
A $\cup$ B = {x:x $\in$ A or x $\in$ B}
 A $\cup$ B A $\cup$ B $\cup$ C
2. Intersection of Sets
The intersection of set A and B is the set of all members that belong to both A and B.
A $\cap$ B = {x:x $\in$ A and x $\in$ B}
 A $\cap$ B A $\cap$ B $\cap$ C
3. Difference of Sets
The difference of sets A and B is the set of all members of set that belongs to set A only.
i.e. A – B = {x:x $\in$ A but x $\notin$ B}
Similarly, B – A = {x:x $\in$ but x $\notin$ A}
 A - B B - A (A $\cup$ B) - C C - (A $\cup$ B)
4. Complement of a set
The complement of a set A is the set of all members of universal set which does not belong to set A.
$\bar{A}$ = U – A = {x:x $\in$ U, x $\notin$ A}
 $\overline{A}$ U $\overline{A \cup B}$ $\overline{A \cup B \cup C}$ $\overline{A \cap B \cap C}$

Cardinality of relations of sets

1. If A and B are the two overlapping sets. U be the universal set. Then,
n(A $\cup$ B) = n(A) + n(B) – n(A $\cap$ B)
Note: When two sets are disjoint,
n(A $\cup$ B) = n(A) + n(B)
 (A $\cup$ B) in overlapping set (A $\cup$ B) in disjoint set
n(U) = n($\overline{A \cup B}$) + n(A $\cup$ B)
n(U) = n(A) + n(B) – n(A $\cap$ B) + n($\overline{A \cap B}$)
n(A $\cup$ B) = no(A) + no(B) + n(A $\cap$ B)
no(A) = n(A) – n(A $\cap$ B)
no(B) = n(B) - n(A $\cap$ B)

2. If A, B, and C are the three overlapping sets then,
(a) n(A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C) – n(A $\cap$ B) – n(B $\cap$ C) – n(A $\cap$ C) + n (A $\cap$ B $\cap$ C)
n(A $\cup$ B $\cup$ C) = no(A) + no(B) + no(C) + no (A $\cap$ B) + no(B $\cap$ C) + no(A $\cap$ C) + n(A $\cap$ B $\cap$ C)
(b) no(C) = n(C) – n(A $\cap$ C) – n(B $\cap$ C) + n(A $\cap$ B $\cap$ C)
(c) no(A) = n(A) – n(A $\cap$ B) – n(A $\cap$ C) + n(A $\cap$ B $\cap$ C)
(d) no(B) = n(B) – n(A $\cap$ B) – n(B $\cap$ C) + n(A $\cap$ B $\cap$ C)
(e) n(A $\cup$ B $\cup$ C) + n($\overline{A \cup B \cup C}$) = n(U)
$\therefore$ n($\overline{A \cup B \cup C}$) = n(U) - n(A $\cup$ B $\cup$ C)
3. When the sets are disjoint the, n(A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C)

Examples

• A, B and C are the subsets of a universal set U. Draw a venn diagram and insert the cardinality of the following information.
n(U) = 45, n(A) = 15, n(B) = 20, n(C) = 16, n(A $\cap$ B) = 4, n(B $\cap$ C) = 3, n(A $\cap$ C) = 5 and n(A $\cap$ B $\cap$ C) = 2.
Solution:
Here, n(U) = 45
n(A) = 15
n(B) = 20
n(C) = 16
n(A $\cap$ B) = 4
n(B $\cap$ C) = 3
n(A $\cap$ C) = 5
n(A $\cap$ B $\cap$ C) = 2

• If n(A) = 48, n(B) = 51, n(C) = 40, n(A $\cap$ B) = 11, n(B $\cap$ C) = 10, n(A $\cap$ C) = 9, n(A $\cap$ B $\cap$ C) = 4 and n(U) = 120, find the value of n(A $\cup$ B $\cup$ C) and n($\overline{A \cup B \cup C}$). Present the above information in venn diagram.
Solution:
Here, n(U) = 120
n(A) = 48
n(B) = 51
n(C) = 40
n(A $\cap$ B) = 11
n(B $\cap$ C) = 10
n(A $\cap$ C) = 9
n(A $\cap$ B $\cap$ C) = 4
n(A $\cup$ B $\cup$ C) = ?
n($\overline{A \cup B \cup C}$) = ?
Now,
n(A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C) – n(A $\cap$ B) – n(B $\cap$ C) – n(A $\cap$ C) + n (A $\cap$ B $\cap$ C)
or, n(A $\cup$ B $\cup$ C) = 48 + 51 + 40 – 11 – 10 – 9 + 4
$\therefore$ n(A $\cup$ B $\cup$ C) = 113
n($\overline{A \cup B \cup C}$) = n(U) - n(A $\cup$ B $\cup$ C)
or, n($\overline{A \cup B \cup C}$) = 120 – 113
$\therefore$ n($\overline{A \cup B \cup C}$) = 7

Solve the following problems:

1. A and B are two subsets of a universal set 'U' in which n(U) = 70, n(A) = 40, n(B) = 20 and n($\overline{A \cup B}$) = 15.
(i) Show the above information in venn diagram.
(ii) Find the value of n(A $\cap$ B).
2. If n(A) = 48, n(B) = 51, n(C) = 40, n(A $\cap$ B) 11, n(B $\cap$ C) = 10, n(C $\cap$ A) = 9, n(A $\cap$ B $\cap$ C) = 4 and n(U) = 120, find the values of n(A $\cup$ B $\cup$ C) and n($\overline{A \cup B \cup C}$). Present the above information in venn diagram.
3. In a survey of a community, 45% of the people like Dashain festival. 65% like Tihar festival and 20% like both festivals.
(i) Show it in venn diagram.
(ii) What percent of them don’t like both.
4. In a class of 55 students, 15 students liked Maths but not English and 18 students liked English but not Maths. If 5 students did not like both how many students like both subjects? Represent the above information in venn diagram.
5. In a survey of 119 students, it was found that 16 drink neither coke nor Pepsi, 69 drink coke and 39 drink pepsi.
(i) How many students drink coke only?
(ii) How many students drink Pepsi only?
(iii) Show the information in venn diagram.
6. In an examination, 40% of the students passed in Mathematics only and 30% passed in Science only. If 10% of the students were failed in both subjects,
(i) What percent of students passed in both subjects?
(ii) What percent of students passed in Mathematics?
(iii) Represent the above information in venn diagram.
7. In a survey of a community, it was found that 55% like summer season, 20% like winter season, 40% don’t like both seasons and 750 like both seasons. By using venn diagram, find the number of people who like winter season.
8. In a survey, one third children like only mango and 22 don’t like mango at all. Also $\frac{2}{5}$ of children like orange but 12 like none of them.
(i) Show the above data in venn diagram.
(ii) How many children like both types of fruit?
9. Among 40 artists, each can sing or dance. If the ratio of the artist who can sing only and dance only is 5:3 and who can sing and dance both is 20% then, find the number of artist who can sing.
10. In a survey of 70 students, 24 liked only two of volleyball, football and basketball. 6 students liked all the three. If 10 didn’t like any of these games, find the number of students who liked only one.
11. In a survey of 200 students, 70 are fond of playing cricket , 80 football and 100 basketball. Similarly, 20 are fond of playing cricket and football only, 30 cricket and basketball only. 40 are fond of playing basketball only. If 20 students are fond of playing football and basketball only, by drawing venn diagram,find:
(i) Number of students who play all games
(ii) How many students don’t play any game?