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).

Answer:

 We know that the formula:  n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )

Here     38 = 17+23- n(X ∩Y)

                n(X∩Y)=17+23-38 = 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?

Answer:  using the formula : n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )

             18=8+15- n(X ∩Y)

so   n(X∩Y)=15+8-18 = 5

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

can speak both Hindi and English?

Answer:

Let H be the set of people who speak Hindi, and E be the set of people who speak English

∴ n(H ∪ E) = 400, n(H) = 250, n(E) = 200

n(H ∩ E) = ?

We know that:

n(H ∪ E) = n(H) + n(E) – n(H ∩ E)

∴ 400 = 250 + 200 – n(H ∩ E)

⇒ 400 = 450 – n(H ∩ E)

⇒ n(H ∩ E) = 450 – 400

∴ n(H ∩ E) = 50

Thus, 50 people can speak both Hindi and English.

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?

Answer:

It is given that:

n(S) = 21, n(T) = 32, n(S ∩ T) = 11

We know that:

n (S ∪ T) = n (S) + n (T) – n (S ∩ T)

∴ n (S ∪ T) = 21 + 32 – 11 = 42

Thus, the set (S ∪ T) has 42 elements.

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?

Answer:

It is given that:

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

We know that:

n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)

∴ 60 = 40 + n(Y) – 10

∴ n(Y) = 60 – (40 – 10) = 30

Thus, the set Y has 30 elements.

6: In a group of 70 people, 37 like coffee, 52 like tea, and each person likes at least one of the

two drinks. How many people like both coffee and tea?

Answer:

Let C denote the set of people who like coffee, and

T denote the set of people who like tea

n(C ∪ T) = 70, n(C) = 37, n(T) = 52

We know that:

n(C ∪ T) = n(C) + n(T) – n(C ∩ T)

∴ 70 = 37 + 52 – n(C ∩ T)

⇒ 70 = 89 – n(C ∩ T)

⇒ n(C ∩ T) = 89 – 70 = 19

Thus, 19 people like 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?

Answer:

Let C denote the set of people who like cricket, and

T denote the set of people who like tennis

∴ n(C ∪ T) = 65, n(C) = 40, n(C ∩ T) = 10

We know that:

n(C ∪ T) = n(C) + n(T) – n(C ∩ T)

∴ 65 = 40 + n(T) – 10

⇒ 65 = 30 + n(T)

⇒ n(T) = 65 – 30 = 35

Therefore, 35 people like tennis.

but,

n(C ∩ T)=10

means 10 people like both

Thus,  people like only tennis = n(T)-n(C ∩ T)=10

8: In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and

French. How many speak at least one of these two languages?

Answer:

Let F be the set of people in the committee who speak French, and S be the set of people in the committee who speak Spanish

∴ n(F) = 50, n(S) = 20, n(S ∩ F) = 10

We know that:

n(S ∪ F) = n(S) + n(F) – n(S ∩ F)

= 20 + 50 – 10

= 70 – 10 = 60

Thus, 60 people in the committee speak at least one of the two languages.