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Set-Builder, Roster Notation, and Classifying Numbers MATH 017 Intermediate Algebra S. Rook Overview • Section 1.2, Objective 2 in the textbook – Set-Builder & Roster Notation – Classifying Numbers 2 Set-Builder & Roster Notation Set-Builder Notation • Set-Builder Notation: describes, but does not explicitly list the elements of a set. • Example: {x | x is an even number}, – The | (vertical bar) is pronounced “such that” • A common exercise is to take a set written in set-builder notation and convert it into what is known as roster notation. 4 Roster Notation • Roster Notation: explicitly listing the elements of a set. – When listing elements, we use set notation and place the elements between and left { and right } (called curly braces). 5 Set-Builder & Roster Notation (Example) Ex 1: Given the set {x | x is an even number between 0 and 10 inclusive}, list its members using roster notation. 6 Set-Builder & Roster Notation (Example) Ex 2: Given the set {x | x is a factor of 5}, list its members using roster notation. 7 Set-Builder & Roster Notation (Example 2 Continued) We use … (ellipses) to denote a set extending infinitely in the same pattern. The set of even numbers can then be expressed as {0, 2, 4, 6, …} The set of odd numbers can then be expressed as {1, 3, 5, 7, 9, 11,…} 8 Set-Builder & Roster Notation (Example) Ex 3: Given the set {x | x is both an odd and even number}, list its members using roster notation. 9 Classifying Numbers Classifying Numbers • Sets of numbers to be familiar with: – Natural numbers (counting numbers): {1, 2, 3, 4, 5,…} – Whole numbers: the natural numbers along with 0. {0, 1, 2, 3, 4,…} – Integers: the natural numbers, opposite of the natural numbers, and zero. {…, -2, -1, 0, 1, 2,…} 11 Classifying Numbers (continued) – Rational numbers: any number that can be expressed as the quotient of two integers, a, b, b ≠ 0. {a/b | a and b are integers, b ≠ 0} – Irrational numbers: any number that CANNOT be expressed as the quotient of two integers. – Real numbers: any number that lies on the number line. 12 Classifying Numbers (continued) • A number can lie in more than one set. – The chart on page 12 in the textbook is helpful. • Presented in a hierarchical manner. • As the chart is traced upwards, the number sets proceed from the specific to the general. – Once a number has been classified as specifically as possible, follow the chart upwards until the real numbers are reached. • Example: 7 can most specifically be classified as a natural number. Following the chart upwards, it can be a whole number, integer, rational number, and a real number. 13 Classifying Numbers (continued) 14 Classifying Numbers (Example) Ex 4: Which sets of numbers does -0.666… belong? 15 Classifying Numbers (Example 4 Continued) • As shown in the example, numbers can be in hidden format. A few to look out for: – Terminating or repeating decimals: are equivalent to fractions hence rational numbers – Square roots: are equivalent to natural numbers if the number under the radical is a perfect square, a whole number if the number under the radical is 0, and an irrational number otherwise. • A common exercise is to be presented with a set of numbers and then asked to classify each member of the set. 16 Classifying Numbers (Example) Ex 5: Given the set S 7, 16 ,0, 1 , 3 , ,0.3,0.3 , 2 list the elements in the following sets: a) b) c) d) e) f) Irrational numbers Whole numbers Integers Natural numbers Rational numbers Real numbers 17 Writing Set-Builder Notation Ex 6: Given the set S = {-3, -2, -1, 0, 1, 2}, express S in set-builder notation – be specific as possible. 18 Writing Set-Builder Notation (Continued) Ex 7: Given the set S = {1, 2, 3, 4, 5}, express S in set-builder notation – be specific as possible. 19 Summary • After studying these slides, you should know how to do the following: – Understand the concepts of roster & set-builder notation – Convert a set expressed in set-builder to roster notation and vice versa – Distinguish between natural numbers, whole numbers, integers, rational numbers, and real numbers – Given a random set of numbers, identify the elements that correspond with at least one classification (see above) • Additional Practice – Attempt the suggested problems found on the course website. 20