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Section 1.3

Section 1.3 . The Language of Sets. Objective. Use three methods to represent sets. Define and recognize the empty set. Use the symbols and . Apply set notation to sets of natural numbers. Determine a set’s cardinal number. Distinguish between finite and infinite sets. Key Terms.

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Section 1.3

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  1. Section 1.3 The Language of Sets

  2. Objective • Use three methods to represent sets. • Define and recognize the empty set. • Use the symbols and . • Apply set notation to sets of natural numbers. • Determine a set’s cardinal number. • Distinguish between finite and infinite sets.

  3. 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.

  4. Three Methods used to Designate Sets • Word Description: describe the set using words. • Roster Form: set of elements listed inside a pair of braces { } separated by commas. • Braces are important because they indicate a set. Never use parenthesis ( ), or brackets [ ]. • 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. • Set-Builder Notation: also called set-generator notation. • A = {x/condition(s)}…This is read as “The set A is the set of all elements x such that certain conditions are met”.

  5. 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}

  6. 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.

  7. 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.

  8. 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. r {a, b, c,…,z} 7 {1, 2, 3, 4, 5} {a} {a, b}

  9. 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.

  10. Example 5: Natural Numbers • Express each of the following sets using roster method. • Set A is the set of natural numbers less than 5. • Set B is the set of natural numbers greater than or equal to 25. • E = {x/xN and x is even}

  11. Example 6: • Express each of the following sets using the roster method. • {x/x N and x < 100} • {x/x N and 70 < x < 100}

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

  13. Example 8: Well-Defined • Well-Defined: the set contents can be clearly defined.

  14. 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 = {7, 9, 11, 13} B = {0} C={13, 14, 15,…,22, 23}

  15. 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}

  16. Section 1.3 Assignments • Classwork: • Textbook pg. 28/2, 4…Set-builder Notation 6, 8…Roster Form10, 12…Word Description 30 – 64 Even • Must write problems and show ALL work to receive credit for the assignment. • Homework:

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