Practice Set 1.4 Sets Class 9th Mathematics Part I MHB Solution

Practice Set 1.4: Sets
Class 9th Mathematics (MHB)

Question 1

If n(A) = 15, n(A ∪ B) = 29, and n(A ∩ B) = 7, then find n(B).

Answer:

We are given the following values:

  • n(A) = 15
  • n(A ∪ B) = 29
  • n(A ∩ B) = 7

We use the formula for the union of two sets:

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Substitute the given values into the formula:

⇒ 29 = 15 + n(B) - 7

Simplify the right side:

⇒ 29 = 8 + n(B)

Solve for n(B):

⇒ n(B) = 29 - 8

⇒ n(B) = 21

Question 2

In a hostel there are 125 students, out of which 80 drink tea, 60 drink coffee and 20 drink tea and coffee both. Find the number of students who do not drink tea or coffee.

Answer:

Let T be the set of students who drink tea, and C be the set of students who drink coffee.

  • Total students in the universal set, n(U) = 125
  • Number of students who drink tea, n(T) = 80
  • Number of students who drink coffee, n(C) = 60
  • Number of students who drink both, n(T ∩ C) = 20

First, find the number of students who drink at least one beverage, n(T ∪ C):

n(T ∪ C) = n(T) + n(C) - n(T ∩ C)

⇒ n(T ∪ C) = 80 + 60 - 20

⇒ n(T ∪ C) = 140 - 20

⇒ n(T ∪ C) = 120

The number of students who do not drink tea or coffee is the total number of students minus those who drink at least one:

⇒ Number = n(U) - n(T ∪ C)

⇒ Number = 125 - 120

Therefore, 5 students do not drink tea or coffee.

U = 125 Tea Coffee 60 20 40 5

Question 3

In a competitive exam, 50 students passed in English, 60 students passed in Mathematics, and 40 students passed in both subjects. None of them failed in both subjects. Find the number of students who passed in at least one of the subjects.

Answer:

Let E be the set of students who passed in English, and M be the set of students who passed in Mathematics.

  • Number of students who passed in English, n(E) = 50
  • Number of students who passed in Mathematics, n(M) = 60
  • Number of students who passed in both, n(E ∩ M) = 40

We need to find the number of students who passed in at least one subject, which is n(E ∪ M).

n(E ∪ M) = n(E) + n(M) - n(E ∩ M)

⇒ n(E ∪ M) = 50 + 60 - 40

⇒ n(E ∪ M) = 110 - 40

⇒ n(E ∪ M) = 70

Question 4

A survey was conducted on 220 students. 130 students informed their hobby is rock climbing and 180 students informed their hobby is sky watching. 110 students follow both hobbies. Answer the following:

  • How many students do not have any of the two hobbies?
  • How many follow the hobby of rock climbing only?
  • How many students follow the hobby of sky watching only?

Answer:

Let R be the set for rock climbing and S for sky watching.

  • Total students, n(U) = 220
  • n(R) = 130
  • n(S) = 180
  • n(R ∩ S) = 110

First, find the number of students with at least one hobby, n(R ∪ S):

n(R ∪ S) = n(R) + n(S) - n(R ∩ S)

⇒ n(R ∪ S) = 130 + 180 - 110 = 200

(i) Students with no hobbies = n(U) - n(R ∪ S) = 220 - 200 = 20.

(ii) Rock climbing only = n(R) - n(R ∩ S) = 130 - 110 = 20.

(iii) Sky watching only = n(S) - n(R ∩ S) = 180 - 110 = 70.

Summary: 20 have no hobby, 20 do only rock climbing, and 70 do only sky watching.

U = 220 Rock Climbing Sky Watching 20 110 70 20

Question 5

Observe the given Venn diagram and write the following sets:

i. A, ii. B, iii. A ∪ B, iv. U, v. A', vi. B', vii. (A ∪ B)'

U A B x, y, z m, n p, q, r s, t

Answer:

Based on the Venn diagram:

  1. A = {x, y, z, m, n}
  2. B = {p, q, r, m, n}
  3. A ∪ B (all elements in A or B or both) = {x, y, z, m, n, p, q, r}
  4. U (the universal set, all elements shown) = {x, y, z, m, n, p, q, r, s, t}
  5. A' (complement of A, elements not in A) = {p, q, r, s, t}
  6. B' (complement of B, elements not in B) = {x, y, z, s, t}
  7. (A ∪ B)' (elements not in the union of A and B) = {s, t}
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