Understanding Set Notation

Students will use mathematical symbols to describe sets.

FHS Algebra Review 2

Warm Up• Classify each of the following

numbers. Write all classifications that apply (real, rational, irrational, integers, whole, natural)

1. -4

2. 3.7

3. π

{Z, Q, R}{Q, R}

{Irrational, R}

FHS Algebra Review 3

What is a Set?

• A set is collection of items called elements. For example we could call the people in this class a set.

• A subset is a set whose elements belong to another set. For example, subsets could be: “boys” , “girls” or “people that are taking band”.

• If a set has no members we say that it is the empty set. In this class that could be people under the age of 10.

FHS Algebra Review 4

Describing a Set• There are several ways to describe a set.

One way is with words. Two other ways use braces.

• In describing a set, braces { } mean “a set of”. Then there are two different ways to describe the set inside the braces:

• Roster notation: lists all the elements of the set, or uses dots to represent the missing members. So we could write the set of whole numbers less than 11 as:

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} or{0, 1, 2, 3, . . ., 10}

FHS Algebra Review 5

Set-Builder Notation• Another way to describe a set is by using

set-builder notation.• This method uses symbols. {x|x < 11 and x∈W}

• The braces { } mean “the set of” • “x|” means “all numbers x such that”• Everything after the “|” describes the set.

In this case it means x is less than 11 and is a whole number.

FHS Algebra Review 6

An Example

1. Describe the set of whole numbers less than 20 but greater than 12 using:

a. Set-Builder notation

b. Roster notation

or

{x|x < 20 and x > 12 and x∈W}

{13, 14, 15, 16, 17, 18, 19}{13, 14, 15, . . .

19}

FHS Algebra Review 7

The Language of Sets• If the elements of a set can be counted, we

say this set is finite.

• If the number of members of a set continues without end, then we say this set is infinite.

• Is the set of all whole numbers less than 100 an infinite or finite set?

• Is the set of all real numbers between 2 and 4 an infinite or finite set?

finite

infinite

FHS Algebra Review 8

How do we graph real numbers?

• Since rational and irrational numbers make up all the other numbers on the number line, we graph the set of real numbers like this:

• We fill in all the space since real numbers represent all the numbers on the number line.

| | | | | | | -2 -1 0 1 2 3 4

FHS Algebra Review 9

Interval Notation• Another way to describe a set of real

numbers (and this should only be used for real numbers) is called interval notation.

• In interval notation, the beginning point and the end point are listed inside parenthesis ( ) or brackets [ ].

• This method is used to describe an interval on a number line.

• If the interval is infinite (goes on forever)

then we use the symbol ∞.

FHS Algebra Review 10

Interval Notation• Here are some examples of interval notation

The description of this set using interval notation is:

The description of this set using interval notation is:

| | | | | | | -2 -1 0 1 2 3 4

| | | | | | | -2 -1 0 1 2 3 4

[-1, 3] - the brackets mean that the numbers are included.

(-1, 2) - the parentheses mean that the numbers are not included.

FHS Algebra Review 11

Interval Notation• Interval notation with an infinite interval

The description of this set using interval notation is:

The description of this set using interval notation is:

| | | | | | | -2 -1 0 1 2 3 4

| | | | | | | -2 -1 0 1 2 3 4

(-∞, 3] The 3 is included but infinity can never be reached - always has a parenthesis.

(2, ∞)