Math 1342 7.2 Sets

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Slide 1

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Chapter 7

Logic, Sets,
and
Counting

Section 2
Sets

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 2

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Set Properties and Set Notation

A set is any collection of objects specified so that we can tell if an object is or is not in the collection.
We use capital letters, such as A, B, and C to designate sets.

Each object in a set is called a member, or element, of the set.

Symbolically,
a ∈ A means “a is an element of set A”
a ∉ A means “a is not an element of set A”

A set without any elements is called the empty, or null, set.

For example, the set of all people 20 feet tall is empty.
Symbolically, ∅ denotes the empty set.

Multiple Choice

$$5\in\left\{9,5\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \$$

• False

• True

Multiple Choice

$$6\in\left\{9,5,2\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \$$

• False

• True

Multiple Choice

$$6\notin\left\{9,5,2\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \$$

• False

• True

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Slide 3

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Set Properties and Set Notation

A set is described either by listing the elements between braces { } (the listing method) or by enclosing a rule within braces that determines the elements of the set (the rule method).

​(the vertical bar is read “such that.”)

If P(x) is a statement about x, then

​S = {x | P(x)} means “S is the set of all x such that P(x) is true”

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 4

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Open Ended

Write the resulting set using the listing method.

{ $$x$$ | $$x^2=25$$ }

Open Ended

Write the resulting set using the listing method.

{ $$x$$ | $$x$$ is an even number between 2 and 10, inclusive}

Multiple Choice

$$12\in\left\{2,4,6,8,...\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \$$

• False

• True

Multiple Choice

$$5\notin\left\{2,4,6,8,...\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \$$

• False

• True

Open Ended

Write the resulting Set using the listing method.

S is the set of all even integers greater than 6.

Dropdown

Consider the set N of positive integers to be the universal set.

{n∊N | n ≥ 98}.

The set is ​

Select answer

because the positive integers starting at 98 can continue increasing​

Select answer

Dropdown

Consider the following set.

{1, 2, 3, 4, 8, 12, 16}.

The set is ​

Select answer

because

Select answer

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 5

Subsets

If each element of a set A is also an element of set B, we say that A is a subset of B.
For example, the set of all women students in a class is a subset of the whole class.
Note that the definition implies that every set is a subset of itself.

If sets A and B have exactly the same elements, the two sets are said to be equal.

Symbolically,
A ⊂ B means “A is a subset of B”

A = B means “A and B have exactly the same elements”

A ⊄ B means “A is not a subset of B”

A ≠ B means “A and B do not have exactly the same elements”

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 7

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Example 2 Set Notation

If A = {–3, –1, 1, 3}, B = {3, –3, 1, –1}, and
C = {–3, –2, –1, 0, 1, 2, 3}, then each of the following
statements is true:

A ⊂ C

A ⊂ B

C ≠ A

C ⊄ A

B ⊂ A

∅ ⊂ A

∅ ⊂ C

∅ ∉ A

​A=B

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Slide 8

Example 3

List all subsets of the set {a, b, c}.

List the set with no elements ∅
List all subsets with exactly one element, {a}, {b}, {c}
List all subsets with exactly two elements, {a, b}, {a, c}, {b, c}
Don’t forget to include the set itself, {a, b, c}

The subsets are {a, b, c}, {a, b}, {a, c}, {b, c}, {a}, {b}, {c},
and ∅.

This set with three elements has eight subsets.

Multiple Choice

Indicate whether the statement is true or false.

{b, h, j, k} = {j, k, h, b}

• True

• False

Dropdown

Indicate whether the statement is true or false.

{0, 6} $$\subset$$ {0, 3, 5, 6}

Select answer

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 9

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Venn Diagrams and Set

Operations

Suppose U is the set of all rental units in a city, and A is the set of all rental units within one mile of the college campus.
It is natural to picture A using Figure 1, called a Venn diagram.

The circle is an imaginary boundary that separates the elements of A (inside the circle) from the elements of U that are not in A (outside the circle).

In the context of the example, the shaded region represents the set of all rental units within one mile of campus. The unshaded region represents all rental units in the city.

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 11

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Venn Diagrams and Set

Operations

Let U be the set of all 26 lowercase letters in the English alphabet, and let A = {a, b, c, d, e}, and let B = {d, e, f, g, h, i}. Figure 2 shows the Venn diagram for sets A and B.

The elements d and e belong to both A and B.

Elements a, b, c belong to A but not to B.

Elements f, g, h, i belong to B but not to A.

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 12

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Venn Diagrams and Set

Operations

Sets can have large numbers of elements. In this case, we generally do not write the names of the elements and we represent the Venn diagram for sets A and B as shown in Figure 3 or Figure 4.

​n(A)=5

n(B)=6

n(U)=26

Math Response

$$n\left(B\right)=$$

Type answer here

Correct Answer: 70

Deg^°

Rad

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 14

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Union of Sets A and B

Definition Union

A ∪ B = {x | x ∈ A or x ∈ B}

We use the word or in the way it is always used in mathematics; that is, x may be an element of set A or set B or both.

The shaded region in Figure 5
illustrates A ∪ B.

​n(A∪B)=9

Math Response

$$n\left(A\cup B\right)=$$

Type answer here

Correct Answer: 150

Deg^°

Rad

Open Ended

Write the union of the given sets.

{2, 7}∪{1, 6, 7}

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 15

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Intersection of Sets A and B

Definition Intersection

A ∩ B = {x | x ∈ A and x ∈ B}

To be in the intersection, x must be an element of both set A
and set B.

The shaded region in Figure 6
illustrates A ∩ B.

Note: A ∩ B ⊂ A and A ∩ B ⊂ B

​n(A∩B)=2

Open Ended

Write the intersection of the given sets.

{2, 7}∩{1, 6, 7}

Math Response

$$n\left(A\cap B\right)=$$

Type answer here

Correct Answer: 20

Deg^°

Rad

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Slide 16

Intersection of Sets A and B

If A ∩ B = ∅, sets A and B are said to be disjoint as shown in
Figure 7.

Disjoint sets have no shared elements.

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Slide 17

Complement of a Set

The complement of set A (relative to a universal set U),
denoted by A′, is the set of elements in U that are not in A.
(Fig. 8)

Definition Complement

A′ = {x ∈ U | x ∉ A}

Open Ended

Let U={2, 3, 4, 5, 6, 7, 9} and A={2, 6, 7}

Find A'

Multiple Choice

Consider D the set of all cards in a standard deck of 52 be the universal set.

A={c ∈ D | c is a heart}

B={c ∈ D | c is a 10}

Are A and B disjoint?

• yes, disjoint

• no, NOT disjoint

Multiple Choice

Consider D the set of all cards in a standard deck of 52 be the universal set.

A={c ∈ D | c is a heart}

B={c ∈ D | c is a 10}

Are A' and B disjoint?

• yes, disjoint

• no, NOT disjoint

Multiple Choice

Consider D the set of all cards in a standard deck of 52 be the universal set.

A={c ∈ D | c is a heart}

B={c ∈ D | c is a diamond}

Are A and B disjoint?

• yes, disjoint

• no, NOT disjoint

Multiple Choice

Consider D the set of all cards in a standard deck of 52 be the universal set.

A={c ∈ D | c is a heart}

B={c ∈ D | c is a diamond}

Are A and B' disjoint?

• yes, disjoint

• no, NOT disjoint

Math Response

$$n\left(A'\right)=$$

Type answer here

Correct Answer: 200

Deg^°

Rad

Math Response

$$n\left(B\cap A'\right)=$$

Type answer here

Correct Answer: 50

Deg^°

Rad

Math Response

$$n\left(B\cup B'\right)=$$

Type answer here

Correct Answer: 300

Deg^°

Rad

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Slide 23

Example 6 Exit Polling

In the 2016 presidential election, an exit poll of 100 voters produced the results in the table (23 men voted for Clinton, 29 women voted for Clinton, 26 men voted for Trump, and 22
women voted for Trump).

Let the universal set U be the set of 100 voters, C the set of voters for Clinton, T the set of voters for Trump, M the set of male voters, and W the set of female voters.

​(A) Find n(C ∩ M), n(C ∩ M′), n(M ∩ C′), and n(C′ ∩ M′).

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 25

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Example 6 Exit Polling

Math Response

Find $$n\left(B\right)=$$

Type answer here

Correct Answer: 87

Deg^°

Rad

Math Response

Find $$n\left(A\cap F\right)=$$

Type answer here

Correct Answer: 19

Deg^°

Rad

Math Response

Find $$n\left(F\cup B\right)=$$

Type answer here

Correct Answer: 106

Deg^°

Rad

Math Response

Find $$n\left(A\cup F\right)=$$

Type answer here

Correct Answer: 99

Deg^°

Rad

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 26

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Example 7 Insurance

Using a random sample of 100 insurance customers, an insurance company generated the Venn diagram in Figure 10 where A is the set of customers who purchased auto
insurance, H is the set of customers who purchased homeowner’s insurance, and L is the set of customers who purchased life insurance.

(A) How many customers purchased
auto insurance?

Solution n(A) = 50 + 16 + 2 + 8 = 76

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 27

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Example 7 Insurance

(B) Use Figure 10 to shade the region H ∪ L and find n(H ∪ L).

n(H ∪ L) = 15 + 16 + 8 + 6 + 2 + 1 = 48
Out of the 100 surveyed customers, 48 purchased homeowner’s insurance or life insurance.

Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Slide 28

Barnett, College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14e

Example 7 Insurance

(C) Use Figure 10 to shade the region A ∩ H ∩ L′ and find n(A ∩ H ∩ L′ ).

n(A ∩ H ∩ L′ ) = 16

Out of the 100 surveyed customers, 16 purchased auto insurance and homeowner’s insurance but did not purchase life insurance.

Match

Match the following

(A∩B'⋂C)'

A'∩B⋂C'

A∩B⋂C

A∩B'⋂C

• (A∩B'⋂C)'

• A'∩B⋂C'

• A∩B⋂C

• A∩B'⋂C

Drag and Drop

For P={3, 9, 13}, Q={1, 6, 11}, and R={4, 6, 9, 11}

P ∪ (Q ∩ R)={​

Placeholder

,​

Placeholder

,​

Placeholder

,​

Placeholder

,​

Placeholder

}

Drag these tiles and drop them in the correct blank above

3

6

9

11

13

1

4