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
-
Winter but not summer
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.