Section 1.3The Language of Sets

Objective1. Use three methods to represent sets.2. Define and recognize the empty set.3. Use the symbols and .4. Apply set notation to sets of natural

numbers.5. Determine a set’s cardinal number.6. Distinguish between finite and infinite

sets.

Key Terms

•Set: a collection of objects.

•Elements/Members: the individual objects in the collection.

•Well-Defined: the set contents can be clearly defined.

•Naming Sets: sets normally denoted by a capital letter. Lower-case letters are used to denote elements in a set.

Three Methods used to Designate Sets

1. Word Description: describe the set using words.

2. Roster Form: set of elements listed inside a pair of braces { } separated by commas.

i. Braces are important because they indicate a set. Never use parenthesis ( ), or brackets [ ].

ii. Ellipses: three dots after the last element in a set, indicates the set continues in the same manner up to the last element or to infinity.

3. Set-Builder Notation: also called set-generator notation.

i. A = {x/condition(s)}…This is read as “The set A is the set of all elements x such that certain conditions are met”.

Example 1: Word Description• {Saturday, Sunday}

• {April, August}

▫ NOTE: when writing sets of numbers, be careful of the words “between” and “inclusive”. Inclusive means all numbers are included; between does not.

• {9, 10, 11, 12,…,25}

Example 2: Roster Form

•The set of the months that have exactly 30 days.

•The set of U. S. coins that have a value less than a dollar.

Example 3: Set-Builder Notation•C is the set of all x such that x is a

carnivorous animal.

•The set of natural numbers less than or equal to 6.

Example 4: and •The symbol, , is read “is an element

of” and indicates membership in a set.

•The symbol, , is read “is not an element of” and indicates object not an element of the set.

Determine whether each statement is true or false.a. r {a, b, c,…,z}

b. 7 {1, 2, 3, 4, 5}

c. {a} {a, b}

Natural Numbers

•The set of counting numbers, starting with 1 and going to infinity is called the “Natural Numbers”.▫Natural numbers are represented by a bold

face N.

Example 5: Natural Numbers

• Express each of the following sets using roster method.

a. Set A is the set of natural numbers less than 5.

b. Set B is the set of natural numbers greater than or equal to 25.

c. E = {x/x N and x is even}

Example 6:•Express each of the following sets using the

roster method.

a. {x/x N and x < 100}

b. {x/x N and 70 < x < 100}

Example 7: Empty Set• Empty Set: also called the null set, is the

set that contains no elements. The empty set is represented by { } or ø.▫The empty set is not represented by {ø}.

This represents the set containing the element ø.

Which of the following is the empty set.a. {x/x is a number less than 3 or greater than

5}

b. {x/x is a number less than 3 and greater than 5}

Example 8: Well-Defined

•Well-Defined: the set contents can be clearly defined.

The set of the days of the week.

The set of the three best songs.

Example 9: Cardinality•Cardinal Number: the number of

elements in a set, also called cardinality.▫Represented by the symbol n(A).

Find the cardinal number of each of the following sets:a. A = {7, 9, 11, 13}

b. B = {0}

c. C = {13, 14, 15,…,22, 23}

Example 10: Finite and Infinite Sets

•A set is finite if it contains no elements or n(A) is a natural number.

•A set whose cardinality is not zero or a natural number is called an infinite set.

•{1, 4, 7,. . . ,16}

•{x/x is a N > 58}

Section 1.3 Assignments

•Classwork:▫Textbook pg. 28/2, 4…Set-builder Notation

6, 8…Roster Form 10, 12…Word Description 30 – 64 Even Must write problems and show ALL work to

receive credit for the assignment.

•Homework: