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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.

Respuesta :

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

Ver imagen Newton9022
batolisis

(a) Names only on list A (only Mathematics): 10

(b) Names only on list B (only English): 15

(c) Names on list A or list B or both: 45

(d) Names on exactly one list (only Mathematics or only English): 25

The question gives the information we need to compute the answers:

(a) Names only on list A (only Mathematics)

List A consists of names in only Mathematics class and in both Mathematics and English class

To get the names only in list A:

[tex]\text{ListA}=\text{(ListA and ListB)} + \text{(ListA and not ListB)}\\30=20+\text{(ListA and not ListB)}\\\text{(ListA and not ListB)}=30-20=10[/tex]

(b) Names only on list B (only English)

List B consists of names in only English class and in both Mathematics and English class

To get the names only in list B:

[tex]\text{ListB}=\text{(ListA and ListB)} + \text{(ListB and not ListA)}\\35=20+\text{(ListB and not ListA)}\\\text{(ListB and not ListA)}=35-20=15[/tex]

(c) Names on list A or list B or both

We want to find the combined number of names in list A and list B, without repetition.

To get the names in list A or list B or both lists, sum the result in (a) and (b) with the names in both list A and list B. That is:

[tex]\text{(ListA or ListB)}=\text{(ListA and not ListB)}+\text{(ListB and not ListA)} +\text{(ListA and ListB)} \\\text{(ListA or ListB)}=10+15+20\\\text{(ListA or ListB)}=45[/tex]

(d) Names on exactly one list (only Mathematics or only English)

We want to find the names that only appear in one list, but never both

To get the names in only list A or only list B, sum the result in (a) and (b). That is:

[tex]\text{(only ListA or only ListB)}=\text{(ListA and not ListB)}+\text{(ListB and not ListA)} \\\text{(only ListA or only ListB)}=10+15\\\text{(only ListA or only ListB)}=25[/tex]

Learn more about sets here: https://brainly.com/question/2166579

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