问题1.如果X和Y是n(X)= 17，n(Y)= 23和n(X∪Y)= 38的两个集合，则求n(X∩Y)。
解决方案：
n (X) = 17
n (Y) = 23
n (X U Y) = 38
So we will write this as :
n (X U Y) = n (X) + n (Y) – n (X ∩ Y)
Putting values,
38 = 17 + 23 – n (X ∩ Y)
So,
n (X ∩ Y) = 40 – 38 = 2
∴ n (X ∩ Y) = 2
问题2.如果X和Y是两组，使得X∪Y有18个元素，则X有8个元素，而Y有15个元素； X∩Y有多少个元素?
解决方案：
n (X U Y) = 18
n (X) = 8
n (Y) = 15
So we will write this as :
n (X U Y) = n (X) + n (Y) – n (X ∩ Y)
Putting values,
18 = 8 + 15 – n (X ∩ Y)
So,
n (X ∩ Y) = 23 – 18 = 5
∴ n (X ∩ Y) = 5
问题3.在一个400人的小组中，有250人会说印地语，而200人会说英语。有多少人会说北印度语和英语?
解决方案：
Let ‘A’ is the set of people who speak Hindi & ‘B’ is the set of people who speak English
Given,
n(A ∪ B) = 400
n(A) = 250
n(B) = 200
So we will write this as :
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
Putting values,
400 = 250 + 200 – n(A ∩ B)
400 = 450 – n(A ∩ B)
So,
n(A ∩ B) = 450 – 400
n(A ∩ B) = 50
∴ 50 people can speak both Hindi & English.
问题4.如果S和T是两个集合，使得S拥有21个元素，T具有32个元素，并且S∩T具有11个元素，那么S∪T有多少个元素?
解决方案：
Given, n(S) = 21
n(T) = 32
n(S ∩ T) = 11
So we will write this as :
n (S ∪ T) = n (S) + n (T) – n (S ∩ T)
Putting values,
n (S ∪ T) = 21 + 32 – 11
So,
n (S ∪ T) = 42
∴ the set (S ∪ T) has 42 elements.
问题5.如果X和Y是两个集合，使得X具有40个元素，X∪Y具有60个元素，X∩Y具有10个元素，那么Y有多少个元素?
解决方案：
Given, n(X) = 40
n(X ∪ Y) = 60
n(X ∩ Y) = 10
So we will write this as :
n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)
Putting values,
60 = 40 + n(Y) – 10
n(Y) = 60 – (40 – 10) = 30
∴ the set Y has 30 elements.
问题6.在一个70人的小组中，37人喜欢喝咖啡，52人喜欢喝茶，每个人都喜欢两种饮料中的至少一种。有多少人喜欢咖啡和茶?
解决方案：
Let ‘A’ is the set of people who like coffee & ‘B’ is the set of people who like tea
Given,
n(C ∪ T) = 70
n(A) = 37
n(B) = 52
So we will write this as :
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
Putting values,
70 = 37 + 52 – n(A ∩ B)
70 = 89 – n(A ∩ B)
So,
n(A ∩ B) = 89 – 70 = 19
∴ 19 people like both coffee and tea.
问题7.在65人的小组中，有40个人喜欢板球，有10个人喜欢板球和网球。有多少只喜欢网球而不喜欢板球?有多少喜欢网球?
解决方案：
Let ‘A’ is the set of people who like cricket & ‘B’ is the set of people who like tennis
Given,
n(A ∪ B) = 65
n(A) = 40
n(A ∩ B) = 10
So we will write this as :
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
Putting values,
65 = 40 + n(B) – 10
65 = 30 + n(B)
So,
n(B) = 65 – 30 = 35
∴ 35 people like tennis.
And we can say,
(B – A) ∪ (B ∩ A) = B
So,
(B – A) ∩ (B ∩ A) = Φ
n (B) = n (B – A) + n (B ∩ A)
Putting values,
35 = n (B – A) + 10
n (B – A) = 35 – 10 = 25
∴ 25 people like only tennis.
问题8.在一个委员会中，有50人说法语，20人说西班牙语，10人说西班牙语和法语。有多少人会说这两种语言中的至少一种?
解决方案：
Let ‘A’ is the set of people in the committee who speak French & ‘B’ is the set of people in the committee who speak Spanish
Given,
n(A) = 50
n(B) = 20
n(B ∩ A) = 10
So we will write this as :
n(B ∪ A) = n(B) + n(A) – n(B ∩ A)
Putting values,
n(B ∪ A) = 20 + 50 – 10
n(B ∪ A) = 70 – 10
n(B ∪ A) = 60
∴ 60 people in the committee speak at least one of these two languages i.e French & Spanish.