Given below are some solved examples in the set theory.

**Example 1: **

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10}, B = (1, 3, 6, 7, 8} and C = {3, 7}, find A ∩ B, A ∪ C, B ∩ A′, B ∩ C′

**Solution: **

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 4, 6, 8, 10}

B = (1, 3, 6, 7, 8}

C = {3, 7}

A ∩ B = {6, 8}

A ∪ C = {2, 3, 4, 6, 7, 8, 10}

B ∩ A′ = {1, 3, 7}

B ∩ C′ = {1, 6, 8}

**Example 2: **

If U = {Pencil, Pen, Eraser, Notebook}, P = {Pencil, Notebook} and Q = {Pen, Eraser}, find P ∪ Q, P ∩ Q, P ∪ Q', P ∩ Q'

**Solution: **

U = {Pencil, Pen, Eraser, Notebook}

P = {Pencil, Notebook}

Q = {Pen, Eraser}

P ∪ Q = {Pencil, Notebook, Pen, Eraser} = U

P ∩ Q = ∅

P ∪ Q' = {Pen, Eraser, Pencil, Notebook}

P ∩ Q' = ∅

**Example 3:**

At a breakfast buffet, 93 people preferred coffee as a beverage, 47 people preferred juice, 25 preferred both coffee and juice. If each person prefers atleast one of the beverages, then how many people visited the buffet?

**Solution: **

Let A be the set of people who prefer coffee and B be the set of people who prefer juice.

n(A) = 93, n(B) = 47, n(A ∩ B) = 25

n(AUB) = n(A) + n(B) - n(A ∩ B)

Plugging-in all the values,

n(AUB) = 93 + 47 - 25

n(AUB) = 115

Hence, the number of people who visited the buffet is 115.

**Example 4: **

In a class of 50 students, 30 speak Spanish, 15 speak both Spanish and English. How many students speak English?

**Solution: **

Let A be the Set of students who speak Spanish and B be the Set of students who speak English.

n(AUB) = 50, n(A) = 30, n(A ∩ B) = 25

n(AUB) = n(A) + n(B) - n(A ∩ B)

Plugging-in all the values,

50 = 30 + n(B) - 15

n(B) = 35

Hence, the number of students who speak English is 35.