Venn Diagram

Venn Diagram 
Venn Diagram : Pictorial representation of sets by means of diagrams is termed as Venn Diagrams.

Elements of Sets : The objects in a set are termed as elements or members of sets.

Let A and B are two sets, such that

A= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

B= { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 }

For this Venn Diagram representation will be :



Where,

A - B = This set has elments which are only in A

B - A = This set has elments which are only in B

A ∩ B is set which has comman elements both from A and B

Also number of elements in A ∪ B is same as number of elements in B ∪ A

So, n(A ∪ B) = n(B ∪ A)

Also, n(A ∩ B) = n(B ∩ A)

From Venn Diagram we can see that   n(A) = n(A-B) + n(A∩B) ...........(a)

Similarly, n(B) = n(B-A) + n(B∩A) ............(b)

Also from Diagram we can write,

n(A∪B) = n(A-B) + n(A∩B) + n(B-A) .............(c)

On adding (a) and (b)

n(A) + n(B) = n(A-B) + n(B-A) + n(A∩B) + n(A∩B)

or n(A) + n(B) - n(A∩B) = n(A-B) + n(B-A) + n(A∩B) ..................(d)

From equation (c) and (d) we can write

n(A∪B) = n(A) + n(B) - n(A∩B)

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Ex. Among 56 people collected in a dinner party, 24 eats non veg food but not veg food and 28 eats non-veg food.

Q(1)= Find out how how many eat veg and non veg both ?

Solution : here n(N∪V) = 56 , n(N-V) = 24 and n(N) = 28

Now n(N) = n(N-V)+ n(N ∩V)

28=24+ n(N ∩V)

So, n(N ∩V) =4 , Hence 4 people eat veg and non veg both

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Q(2) Find out how many of them eat Veg but not non veg ?

Solution : We can write n(N∪V) = n(N) + n(V) - n(N∩V)

56 = 28 + n(V) - 4

n(V) = 32

Also, n(V) = n(V-N) + n(V∩N)

32 = n(V-N) + 4

n(V-N) = 28, Hence 28 people eats Veg but not non veg

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Ex. In a club of 48 people, 24 plays cricket and 16 plays cricket but not hockey. Find the number of people in club who plays hockey but not cricket ?

Solution :Let C denotes the cricket and H denotes hockey, according to question,

n(C∪H)=48,     n(C)=24     n(C-H)=16

   Now n(C)= n(C - H) + n(C∩H)

   24 = 16 + n(C∩H)

  n(C∩H)= 8

   Now, n(C∪H)=n(C)+n(H)-n(C∩H)

   48 = 24 + n(H) - 8

   n(H)=32

   n(H)=n(H-C)+ n (H ∩C)

   32 = n(H-C) + 8

   n(H-C)= 32-8 = 24

So, people in club who plays hockey but not cricket are 24

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Ex. In a society of 80 people, 42 read Times Of India and 35 read The Hindu, while 8 people don not read any of the two news papers.

Q(1) Find the number of people , who read at least one of the two news papers .

Solution : Here total number of people are 80 out of which 8 do not read any news paper, so 80 - 8 = 72 people read remaning two news papers

So, n(T∪H)=72,   n(T)=42,   n(H)=35

So, the number of people , who read at least one of the two news papers = n(T∪H)=72


Q(2) Find the number of people in society , who read both news papers .

Solution :n(T∪H) = n(T) + n(H) - n(T∩H)

72 = 42 + 35 - n(T∩H)

n(T∩H) = 77 - 72 = 5

So, the number of people in society , who read both news papers = 5

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Ex. In a society 50 % people read Times Of India, 25 % read The Hindu. 20 % read both news papers. What % of people read neither Times Of India nor The Hindu ?

Solution : n(T)=50,   n(H)=25,   n(T∩H)=20

n(T∪H) = n(T) + n(H) - n(T∩H)

n(T∪H) = 50 + 25 - 20 = 55

Since 55 % people read either Times Of India or The Hindu , so remaning 100 - 55 = 45 %

So, 45 % of people read neither Times Of India nor The Hindu