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Learning Objectives for Section 7.2 Sets

After today’s lesson, you should be able to Identify and use set properties and set notation. Perform set operations. Solve applications involving sets.

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Set Properties and Set Notation

Definition: A set is any collection of objects Notation:

e A means “e is an element of A”,

or “e belongs to set A”.

e A means “e is not an element of A”.

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Set Notation (continued)

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”

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Set Properties and Set Notation(continued)

Example of a set: Let A be the set of all the letters in the alphabet. We write that as

A = { a, b, c, d, e, …, z}.

This is called the listing method of specifying a set. We use capital letters to represent sets. We list the elements of the set within braces, separated by

commas. The three dots (…) indicate that the pattern continues.

Question: Is 3 a member of the set A?

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Set-Builder Notation

Sometimes it is convenient to represent sets using set-builder notation.

Example: Using set-builder notation, write the letters of the alphabet.

A = {x | x is a letter of the English alphabet}

This is read, “the set of all x such that x is a letter of the English alphabet.”

It is equivalent to A = {a , b, c, d, e, …, z}

Note: {x | x2 = 9} = {3, -3}This is read as “the set of all x such that the square of x equals 9.”

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Null Set

Example: What are the real number solutions of the equation

x2 + 1 = 0?

Answer: ________________________________________

________________________________________________

We represent the solution as the __________, written ____ or ___.

It is also called the _______________ set.

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Subsets

A is a subset of B if every element of A is also contained in B. This is written

A B.

For example, the set of integers { …-3, -2, -1, 0, 1, 2, 3, …}

is a subset of the set of real numbers.

Formal Definition: A B means “if x A, then x B.”

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Subsets(continued)

Note: Every set is a subset of itself.

Ø (the null set) is a subset of every set.

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Number of Subsets

Example: List all the subsets of set A = {bird, cat, dog}

For convenience, we will use the notation A = {b, c, d} to represent set A.

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Union of Sets(OR)

A B = { x | x A or x B}

In the Venn diagram on the left, the union of A and B is the entire region shaded.

The union of two sets A and B is the set of all elements formed by combining all the elements of set A and all the elements of set B into one set. It is written A B.

AA BB

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Intersection of Sets(AND)

A B = { x | x A and x B}

The intersection of two sets A and B is the set of all elements that are common to both A and B. It is written A B.

AA BB In the Venn diagram on the left, the intersection of A and B is the shaded region.

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Example

Example: Given A = {3, 6, 9, 12, 15} and B = {1, 4, 9, 16} find:

a) A B .

b) A B.

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Disjoint Sets

If two sets have no elements in common, they are said to be disjoint. Two sets A and B are disjoint if

A B = .

Example: The rational and irrational numbers are disjoint.

In symbols:

{ is a rational number}

{ is a irrational number}

Q x x

I x x

Q I

Q I

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The Universal Set

The set of all elements under consideration is called the universal set U.

U

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The Complement of a Set(NOT)

The complement of a set A is defined as the set of elements that are contained in U, the universal set, but not contained in set A. The symbolism and notation for the complement of set A are

}{' AxUxA

In the Venn diagram on the left, the rectangle represents the universal set.

A is the shaded area outside the set A.

U

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Venn Diagram

Refer to the Venn diagram below. The indicated values represent the number of elements in each region. How many elements are in each of the indicated sets?

A B

U

65 12 40

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2) 'n A

3) n A B

4) 'n A B

5) 'n A B

1) n U

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Application*

A marketing survey of 1,000 car commuters found that 600 listen to the news, 500 listen to music, and 300 listen to both.

Let N = set of commuters in the sample who listen to news Let M = set of commuters in the sample who listen to music

Find the number of commuters in the set

The number of elements in a set A is denoted by n(A), so in this case we are looking for

'N M

( ')n N M

The set N (news listeners) consists of

600 elements all together. The middle

part has _______, so the other part must

have _______ elements. Therefore,

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Solution(continued)

' ________ .n N M

Fill in the remaining blanks.

The study is based on 1000 commuters, so n(U)=___________.

______ listen to news but not music.

______ listen to music but not news

NM

U_____ people listen to neither news nor music

_______ listen to both music and news

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Examples From the Text

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