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Problems on Intersection of three Sets

In this section, we will go through some practical problems on intersection of three sets.

For solving such a problems we have to consider the following rules :

If A, B and C are three finite sets then :
1) n ( A ∪ B ∪ C ) =
n(A) + n(B) + n(C) – n ( A ∩ B ) – n(B ∩ C) – n (A ∩ C) + n( A ∩ B ∩ C )

2) n[ A ∩ ( B ∪ C) ] = n ( A ∩ B ) + n ( A ∩ C) – n( A ∩ B ∩ C)

Problems on Intersection of three Sets :

1) In a survey of 200 students of a school it was found that 120 study mathematics, 90 study physics and 70 study chemistry, 40 study mathematics and physics, 30 study physics and chemistry, 50 study chemistry and mathematics and 20 study none of these subjects. Find the number of students who study all three subjects.

Solution :
M = Mathematics ; P = Physics and C = Chemistry

n(M) = 120 n(P) = 90 n (C) = 70 n ( M ∩ P) = 40
n ( P ∩ C ) = 30 n ( C ∩ M ) = 50 n ( M ∪ P ∪ C )’ = 20

Now n(M ∪ P ∪ C)’ = n(U) – n(M ∪ P ∪ C)

20 = 200 – n (M ∪ P ∪ C)

Therefore, n(M ∪ P ∪ C) = 200 – 20 = 180

n(M ∪ P ∪ C)
= n(M) + n(P) + n(C) – n(M ∩ P) – n(P ∩ C) – n(C ∩ M) + n(M ∩ P ∩ C)

180 = 120 + 90 + 70 - 40 - 30 - 50 + n(M ∩ P ∩ C)

⇒ n(M ∩ P ∩ C) =180 - 120 - 90 - 70 + 40 + 30 + 50

⇒ n(M ∩ P ∩ C) = 20.

2) In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find the number of people who read at least one of the newspapers.(problems on intersection of three sets)

Solution :
H = People who read newspaper H.

I = People who read newspaper I.

T = People who read newspaper T.

n(H) = 25 ; n(T)= 26 ; n(I)= 26 ; n(H ∩ I) = 9
n(H ∩ T) = 11 ; n(T ∩ I) = 8 and n(H ∩ T ∩ I) = 3

n(H ∪T ∪ I )= Number of people who read at least one of the newspapers

= n(H) + n(T) + n(I) – n(H ∩ T) – n(T ∩ I) – n(H ∩ I) + n(H ∩ T ∩ I)

n(H ∪T ∪ I )= 25+ 26 + 26 - 11 - 9 - 8 + 3

= 77 - 28 + 3

= 80 - 28

= 52

Hence the number of people who read at least one of the newspapers is 52.

Set Theory

Sets
Representation of Set
Cardinal Number
Types of Sets
Pairs of Sets
Subset
Complement of Set
Union of the Sets
Intersection of Sets
Operations on Sets
De Morgan's Law
Venn Diagrams
Venn-diagrams for sets
Venn-diagrams for different situations
Problems on Intersection of Two Sets
Problems on Intersection of Three Sets
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