Solution:

Using the formula: n (A ∪ B) = n (A) + n (B) − n (A ∩ B)

29 = 15 + n (B) − 7

29 = 8 + n (B)

29 − 8 = n (B)

21 = n (B)

i.e. n (B) = 21

Solution:

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

Now,
n (T) = 80, n (C) = 60 and n (T ∩ C) = 20, n (T ∪ C) = ?

Using the formula:
 n (T ∪ C) = n (T) + n (C) − n (T ∩ C)
∴ n (T ∪ C) = 80 + 60 − 20
∴ n (T ∪ C) = 120 ... (i)
120 students drink either tea or coffee or both.

But, there are 125 students in the hostel.

∴ The number of students who drink neither tea nor coffee
= Total number of students − n (T ∪ C)
= 125 − 120
= 5

∴ 5 students drink neither tea nor coffee.

Solution:

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

Now,
n (E) = 50, n (M) = 60 and n (E ∩ M) = 40, n (E ∪ M) = ?

Using the formula:
 n (E ∪ M) = n (E) + n (M) − n (E ∩ M)
∴ n (E ∪ M) = 50 + 60 − 40
∴ n (E ∪ M) = 70 ... (i)
∴ The number of students who passed in at least one of the subjects is 70.

Solution:

Let, R be the set of students interested in rock climbing and S be the set of students interested in sky watching.

Now,
n (U) = 220, n (R) = 130, n (S) = 180 and n (R ∩ S) = 110, n (R ∪ S) = ?

Using the formula:
 n (R ∪ S) = n (R) + n (S) − n (R ∩ S)
∴ n (R ∪ S) = 130 + 180 − 110
∴ n (R ∪ S) = 200 ... (i)
∴ The number of students who follow at least one of the hobbies is 200.

Therefore, the number of students who do not have any of the two hobbies is:
n (U) − n (R ∪ S) = 220 − 200 = 20 ... (ii)

The number of students who follow the hobby of rock climbing only is:
n (R) − n (R ∩ S) = 130 − 110 = 20 ... (iii)

The number of students who follow the hobby of sky watching only is:
n (S) − n (R ∩ S) = 180 − 110 = 70 ... (iv)

This is better understood using a Venn Diagram which is given below:

Solution:

1. A = {x, y, z, m, n}
2. B = {p, q, r, m, n}
3. A ∪ B = {x, y, z, m, n, p, q, r}
4. U = {x, y, z, m, n, p, q, r, s, t}
5. A' = {p, q, r, s, t}
6. B' = {x, y, z, s, t}
7. (A ∪ B)' = {s, t}