# 0.11 Sets and counting: homework

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This chapter covers principles of sets and counting. After completing this chapter students should be able to: use set theory and venn diagrams to solve counting problems; use the multiplication axiom to solve counting problems; use permutations to solve counting problems; use combinations to solve counting problems; and use the binomial theorem to expand x+y^n.

## Sets and counting

Find the indicated sets.

List all subsets of the following set.

$\left\{\text{Al, Bob}\right\}$
$\left\{\text{Al, Bob}\right\},\left\{\text{Al}\right\},\left\{\text{Bob}\right\},Ø$

List all subsets of the following set.

$\left\{\text{Al, Bob, Chris}\right\}$

List the elements of the following set.

$\left\{\text{Al, Bob, Chris, Dave}\right\}\cap \left\{\text{Bob, Chris, Dave, Ed}\right\}$
$\left\{\text{Bob, Chris, Dave}\right\}$

List the elements of the following set.

$\left\{\text{Al, Bob, Chris, Dave}\right\}\cup \left\{\text{Bob, Chris, Dave, Ed}\right\}$

In Problems 5 - 8, let $\text{Universal set}=U=\left\{a,b,c,d,e,f,g,h,i,j\right\}$ , $V=\left\{a,e,i,f,h\right\}$ , and $W=\left\{a,c,e,g,i\right\}$ .

List the members of the following sets.

$V\cup W$
$\left\{a,e,i,f,h,c,g\right\}$
$V\cap W$
$\overline{V\cup W}$
$\left\{b,d,j\right\}$
$\stackrel{ˉ}{V}\cap \stackrel{ˉ}{W}$

In 9 - 12, let $\text{Universal set}=U=\left\{1,2,3,4,5,6,7,8,9,\mathrm{10}\right\}$ , $A=\left\{1,2,3,4,5\right\}$ , $B=\left\{1,3,4,6\right\}$ , and $C=\left\{2,4,6\right\}$ .

List the members of the following sets.

$A\cup B$
$\left\{1,2,3,4,5,6\right\}$
$A\cap C$
$\overline{A\cup B}\cap C$

Ø

$\stackrel{ˉ}{A}\cup \overline{B\cap C}$

Find the number of elements in the following sets.

In Mrs. Yamamoto's class of 35 students, 12 students are taking history, 18 are taking English, and 4 are taking both. Draw a Venn diagram and determine how many students are taking neither history nor English?

9 students

In the County of Santa Clara 700,000 people read the San Jose Mercury News, 400,000 people read the San Francisco Examiner, and 100,000 read both newspapers. How many read either the Mercury News or the Examiner?

A survey of athletes revealed that for their minor aches and pains, 30 used aspirin, 50 used ibuprofen, and 15 used both. How many athletes were surveyed?

65

In a survey of computer users, it was found that 50 use HP printers, 30 use IBM printers, 20 use Apple printers, 13 use HP and IBM, 9 use HP and Apple, 7 use IBM and Apple, and 3 use all three. How many use at least one of these Brands?

This quarter, a survey of 100 students at De Anza college finds that 50 take math, 40 take English, and 30 take history. Of these 15 take English and math, 10 take English and history, 10 take math and history, and 5 take all three subjects. Draw a Venn diagram and determine the following.

1. The number of students taking math but not the other two subjects.
2. The number of students taking English or math but not history.
3. The number of students taking none of these subjects.
1. 30
2. 60
3. 10

In a survey of investors it was found that 100 invested in stocks, 60 in mutual funds, and 50 in bonds. Of these, 35 invested in stocks and mutual funds, 30 in mutual funds and bonds, 28 in stocks and bonds, and 20 in all three. Determine the following.

1. The number of investors that participated in the survey.

2. How many invested in stocks or mutual funds but not in bonds?

3. How many invested in exactly one type of investment?

## Tree diagrams and the multiplication axiom

Do the following problems using a tree diagram or the multiplcation axiom.

A man has 3 shirts, and 2 pairs of pants. Use a tree diagram to determine the number of possible outfits.

6

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