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Section 1.3 The Language of Sets

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Objective 1.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.

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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.

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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”.

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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}

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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.

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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.

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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}

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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.

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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}

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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}

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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}

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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.

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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}

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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}

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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:

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