第1章布景-第1章的其他练习|套装1
问题11：设A和B。如果对于某些集合X，A∩X = B X = φ且A∪X = B∪X，则表明A = B
解决方案：
A = A ∩ (A ∪ X) //absorption law
A = A ∩ (B ∪ X) //given that A ∩ X = B ∩ X
= (A ∩ B) ∪ (A ∩ X) //distributive law
= (A ∩ B) ∪ φ //given that A ∩ X = φ
A = (A ∩ B) …(1)
Repeating above process by taking B = B ∩ (B ∪ X)
We get B = (A ∩ B) …(2)
from (1) and (2)
A = B
问题12：找到集合A，B和C，使得A∩B，B∩C和A∩C是非空集，并且A∩B∩C = φ
解决方案：
(A ∩ B) should be non-empty means atleast one element is common in between them
… same for (B ∩ C) & (A ∩ C)
A ∩ B ∩ C = φ means there should not be any element common in all the three sets A, B and C
Let A = {1,2} B = {2,3} and C = {1,3}
A ∩ B = {2}
B ∩ C = {3}
A ∩ C = {1}
and A ∩ B ∩ C = φ
问题13：在对一所学校600名学生的调查中，发现150名学生正在喝茶，225名正在喝咖啡，100名同时喝茶和咖啡。找出有多少学生既不喝茶也不喝咖啡?
解决方案：
There are total 600 students
Let A and B represents sets of students taking tea and coffee respectively
n(A) = 150
n(B) = 225
Students taking both tea and coffee = n(A ∩ B) = 100
Students taking either tea or coffee = n(A ∪ B) = ?
By principle of inclusion-exclusion,
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 150 + 225 – 100
= 275
Now,
Number of students who neither take tea nor coffee = total students – Number of Students who either take tea or coffee
= 600 – 275
= 325
There are 325 students who neither take tea nor coffee
问题14：在一组学生中，有100名学生知道印地语，有50名学生知道英语，有25名学生都知道。每个学生都知道印地语或英语。小组中有多少个学生?
解决方案：
Let A and B represents sets of students who knows Hindi and English respectively
n(A) = 100
n(B) = 50
Number of students who know both languages = n(A ∩ B) = 25
It is given that each student knows either Hindi or English
Hence, Number of Students in a group = n(A ∪ B)
By principle of inclusion-exclusion,
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 100 + 50 – 25
= 125
Total students in the group are 125
问题15：在对60人的调查中，发现25人阅读报纸H，26阅读报纸T，26阅读报纸I，9阅读H和I，11阅读H和T，8阅读T和I。 ，3阅读所有三份报纸。找：
(i)至少读过一份报纸的人数。
(ii)准确阅读一份报纸的人数
解决方案：
(i) Total number of people in survey = 60
Let H, I, T represents sets of people reading newspaper H, I and T respectively
n(H) = 25
n(I) = 26
n(T) = 26
People reading both H and I = n(H ∩ I) = 9
People reading both H and T = n(H ∩ T) = 11
People reading both T and I = n(T ∩ I) = 8
People reading all the three newspapers = n(H ∩ I ∩ T) = 3
(i) Number of people who read at least one of the newspapers is given by n(H ∪ I ∪ T)
By principle of inclusion-exclusion,
n(H ∪ I ∪ T) = n(H) + n(I) + n(T) – n(H ∩ I) – n(H ∩ T) – n(T ∩ I) + n(H ∩ I ∩ T)
= 25 + 26 + 26 – 9 – 11 – 8 + 3
= 52
Number of people who read at least one of the newspapers is 52
(ii) Take a look at following Venn diagram
Number of people who read exactly one newspaper are represented by green color in above diagram
= n(H ∪ I ∪ T) – n(H ∩ I) – n(H ∩ T) – n(T ∩ I) + (2 x n(H ∩ I ∩ T))
= 52 – 9 – 11 – 8 + (2 x 3)
= 24 + 6
= 30
Number of people who read exactly one newspaper are 30
问题16：在调查中发现，有21个人喜欢产品A，有26个人喜欢产品B，有29个人喜欢产品C。如果有14个人喜欢产品A和B，则有12个人喜欢产品C和A，有14个人喜欢产品B和C。和8喜欢这三种产品。仅查找有多少喜欢的产品C。
解决方案：
Let A, B, C represents sets of people who liked product A, B and C respectively
n(A) = 21
n(B) = 26
n(C) = 29
People who liked product A and B both = n(A ∩ B) = 14
People who liked product A and C both = n(A ∩ C) = 12
People who liked product B and C both = n(B ∩ C) = 14
People who liked all the three products = n(A ∩ B ∩ C) = 8
Take a look at following Venn diagram
Number of people who only like product C are represented by green colour
= n(C) – n(A ∩ C) – n(B ∩ C) + n(A ∩ B ∩ C)
= 29 – 12 – 14 + 8
= 11
Number of people who only like product C are 11