Exercise- 1.6

1. If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩ Y).

n(X ∪ Y) = 38, n(X) = 17, n(Y) = 23 n(X ∩ Y) = ?

By using the formula:

n(X ∪ Y) = n(X) + n(Y) – n(X – Y)

We find that:

n(X ∩ Y)       =         n(X) + n(Y) – n(X ∪ Y)

n(X ∩ Y)       =         17 + 23 - 38

n(X ∩ Y)       =         2

2. If X and Y are two sets such that X ∪ Y has 18 elements, X has 8 elements and Y has 15 elements, how many elements does X ∩ Y have?

X ∪ Y = 18, X = 18, Y = 15

By using formula:

n(X ∩ Y)       =         n(X) + n(Y) – n(X ∪ Y)

n(X ∩ Y)       =         8 + 15 - 18

n(X ∩ Y)       =         5

3. In a group of 400 people, 250 can speak Hindi, 200 can speak English. How many people can have both English and Hindi.

A = Hindi,   B = English

n(A) = 250,            n(B) = 200

⟹ n(A ∪ B) = 400

n(A ∩ B) = ?

Then,

n(A ∪ B) = n(A) + n(b) – n(A ∩ B)

400 = 250 + 200 – n(A ∩ B)

n(A ∩ B) = 450 - 400owH

 

n(A ∩ B) = 50

4. If S and T are two sets such that S has 21 elements, T has 32 elements and S ∩ T has 11 elements, how many elements does S ∪ T have?

n(S ∩ T) = 11, n(S) = 21, n(T) = 32

By using the formula:

n(S ∪ T) = n(s) + n(T) – n(S ∩ T)

n(S ∪ T) = 21 + 32 -11

n(S ∪ T) = 42

5. If X and Y are two sets such that X has 40 elements X ∪ Y has 60 elements and X ∩ Y has 10 elements, how many elements does Y have?

n(X ∩ Y) = 10, n(X) = 40, n(X ∪ Y) = 60, n(Y) = ?

By using the formula:

n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)

60 = 40 + n(Y) - 10

60 - 30 = n(Y)

n(Y) = 30 elements.

6. In a group of 70 people, 37 like coffee, 52 like tea and each persons like at least one of the two drinks. How many like both coffee and tea?

(A) Coffee, n(A) = 37

(B) Tea, n(B) = 52

n(A ∪ B) = 70, n(A ∩ B) = ?

By using formula:

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

70 = 37 + 52 - n(A ∩ B)

70 = 84 - n(A ∩ B)

n(A ∩ B) = 19

∴ 19 likes both coffee and tea.

7. In a group of 65 people 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket. How many like tennis?

Total People = 65.

People like cricket = 40.

n(C ∩ T) = 10

By using formula:

n(C ∪ T) = n(C) + n(T) + n(C ∩ T)

65 = 40 + n(T) – 10

65 – 30 = n(T)

n(T) = 35.

∴ No. of people who like tennis is 35

∴ No. of people who like only tennis

= n(T) – n(C ∩ T)

= 35 – 10 = 25.

8. In a committee, 50 people speak French, 20 people speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages.

A = For French.

B = for Spanish.

n(A) = 50, n(B) = 20, n(A ∩ B) = 10, n(A ∪ B) = ?

By using formula:

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

n(A ∪ B) = 50 + 20 - 10

n(A ∪ B) = 60

∴ 60 people speak at least one of these two languages