Suppose a list A contains the 30 students in a mathematics class, and a list B contains the 35 students in an English class, and
suppose there are 20 names on both lists. Find the number of students: (a) only on list A, (b) only on list B, (c) on list A or B (or both), (d) on exactly one list.
a. List A has 30 names and 20 are on list B; hence 30 - 20 =10 names are only on list A.
b. Similarly, 35 -20 =15 are only on list B.
c. We seek n(A U B). By inclusion—exclusion,
n(AU B) = n(A) +n(B)- n(A⋂ B) = 30+35 -20 —45.
In other words, we combine the two lists and then cross out the 20 names which appear twice.
d. By (a) and (b), 10 + 15= 25 names are only on one list; that is, n(A⊕ B)= 25.
a. List A has 30 names and 20 are on list B; hence 30 - 20 =10 names are only on list A.
b. Similarly, 35 -20 =15 are only on list B.
c. We seek n(A U B). By inclusion—exclusion,
n(AU B) = n(A) +n(B)- n(A⋂ B) = 30+35 -20 —45.
In other words, we combine the two lists and then cross out the 20 names which appear twice.
d. By (a) and (b), 10 + 15= 25 names are only on one list; that is, n(A⊕ B)= 25.
1 answer:
Answer:
(a)10
(b)15
(c)45
(d)25
Step-by-step explanation:
You can use the attached Venn diagram for illustration.
n(A)=30
n(B)=35
n(A⋂B)=20
(a) Studentsbonly on list A
n(A only)= n(A)-n(A⋂B)=30-20=10
(b)Students only on list B
n(B only)= n(B)-n(A⋂B)=35-20=15
(c)Students on list A or B (or both)
This is the total number of students in the class.
n(AUB)
= n(A only) +n(B only)+ n(A⋂ B)
= 10+15 + 20 = 45
(d)Students on exactly one list.
n(on exactly one list)
= n(A only) +n(B only)
= 10+15
=25
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