Set and Venn Diagram, Solved word problems on Venn diagram, set theory, Mathematics

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 Venn diagram is a graphical representation of sets. Or a pictorial representation of sets represented by the closed figure is called a set diagram or Venn Diagram. Venn diagrams are used to illustrate various operations like union, intersection, and difference, the complement of a set. We can explain the relationship among sets through this in a more significant way.

Venn diagram consists of:

  • A universal set by using a rectangle.
  • A set or subset of a universal set is represented by using a circle or ovals.
Examples
Venn diagram for set union:



Venn diagram for set intersection:


Venn diagram for complement of set A:
Venn diagram for the complement of set B

Example:1
    In a survey of students, 50 students like Mathematics, 30 people like English, 10 people likes both           English and Mathematics. find the total number of students in the survey?
solution:
 Given data,
     Student who like Mathematics = n(M)=50
     Student who like English =n(E)= 30
     The student who like Mathematics and English       both=n(M∩E)=10
To find:
      Total numbers of students =n(M∪E)=?
Formula:
       n(M∪E)=n(M)+n(E)-n(M∩E)
By putting all values:
       n(M∪E)=50+30-10
       n(M∪E)=70
The total number of students present in the survey is 70
Example 2:
In a survey of 100 students it was found that 28 read newspaper A, 30 read newspaper B, 42 read newspaper C, 9 read A &B, 11 read A & C, 6 read B & C and 4 read all the newspaper. How many read newspaper C only?
solution:

Given data,

total number of students=100

n(A) =28

n(B)=30

n(C) =42

n(A⋂B) =9

n(A⋂C)= 11

n(B⋂C) =6

n(A⋂B⋂C) =4

by using venn diagram:

n(only C)= n(C) - n(A⋂C) -n(B⋂C) +n(A⋂B⋂C)

n(only C)=42–11- 6+4

n(only C)=29

the total number of students who read only newspaper is 29.

 Example 3:
A survey asks 200 people “What beverage do you drink in the morning”, and offers choices:
Tea only
Coffee only
Both coffee and tea
Suppose 20 report tea only, 80 report coffee only, 40 reports both. How many people drink tea in the
morning? How many people drink neither tea or coffee?
solution:
this problem can be solved by Venn diagram 
Venn diagram for the given problem is:







number of people who drink neither tea nor coffee =200 - 80 - 20 - 40 = 60
 
Example 4:
A survey asks: “Which online services have you used in the last month?”
Twitter
Facebook
Have used both
The results show 40% of those surveyed have used Twitter, 70% have used Facebook, and 20% have used both. How many people have used neither Twitter or Facebook?
solution:
Venn diagram:

total number of people who either use Facebook or twitter = 70% + 40% - 20 = 90%
total number of people who neither use Facebook nor twitter =100% - 90% = 10%






if you have my query related to set theory you can comment below.






   

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