Example 1:

In a survey, 150 people liked winter, 200 liked summer and 50 liked both summer and winter. Find the number of people who liked

1. Winter but not summer

I did not understand this questions solution

A = (A − B) ∪ (A ∩ B)

n (A) = n (A − B) + n (A ∩ B)

n (A − B) = n (A) − n (A ∩ B)

  = 150 − 50

  = 100

Thus, the number of people who like winter but not summer is 100.

Let A denote the set of people who like winter and B denote the set of people who like summer.

n (A) = 150 n (B) = 200 n (A ∩ B) = 50

Let us find the set whose value we have to find by looking at the following venn-diagram.

From the Venn diagram, we can observe that set A is the union of set (A – B) and the set (A ∩ B).

∴ A = (A − B) ∪ (A ∩ B) 

[We can verify this property by taking any arbitrary element x.

 Let x ∈ (A − B) ∪ (A ∩ B) 

⇒ (x ∈ A and  x  B) or (x ∈ A and  x ∈ B)

⇒ x ∈ A and (x  B or x ∈ B)

⇒ x ∈ A and U

⇒ x ∈ A ∩ U

⇒ x ∈ A]

Thus, n (A) = n (A − B) + n (A ∩ B)

⇒ n (A − B) = n (A) − n (A ∩ B)

  = 150 − 50

  = 100

Thus, the number of people who like winter but not summer is 100.