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Set Theory Using Mathematics to Classify Objects 2.1 The Language of Sets Objectives: • Define sets • Specify sets using both listing (roster method) and set-builder notation • Understand when sets are well-defined • Use the element symbol property • Find the cardinal number of sets • A set is collection of objects. Each object in a set is called an element (member)of the set. • Elements of a set may not share the same characteristics. In this case we represent the set by roster method which means that we list all the elements of the set. • If the elements of a set share the same characteristics, then we can represent the set by either the roster method or represent it by using the so called “set builder notation” • • • • • • • • • Examples: Roster method A={2, 5, dog, ice cream, pen, 2/3, chair} B={-3.7, -1, 0, 1, 2.5, 4.8, 5} C={4, 1, 80, 26, 9, 10, 14, -3} D={4, {1,3,-5},{dog, cat, 7}, Arkansas} Here are some well known stes: N ={1, 2, 3, 4, 5, . . .}= set of natural numbers W={0, 1, 2, 3, 4, 5, . . .}= set of whole numbers I= {. . . , -2, -1, 0, 1, 2, . . . }= set of all integers • Set-builder notation: • • • • Examples: A={x: x is a student in this class} B={x: x is a male person shorter than 5 feet} C={x: x is an African American female in this class} • D={x: x is a number less than 100 and divisible by 3} • P={x: x is a prime number} Well-defined Sets • A set is well-defined if we are able to tell whether or not any particular object is an element of the set. • Example: Which sets are well-defined? (a) { x : x is an Academy Award winner } (b) { x : x is tall } (c) {x : x eats too much} (d) {x : x is a even number} Empty Sets Examples: A = { x : x is a negative natural number } B = { x : x is a pink elephant living in Royer } C={x: x is a person with three eyes} A, B and C the same. Why? • Do and {} mean the same thing? – is the empty set – a set with no members – {} is a set with a member object, namely, the empty set Universal Set • Example: Consider female consumers living in the U.S. The universal set is U x : x is a fem ale cosum er living in the U .S . • Example: Consider the set of Natural numbers. The universal set is U = { 1, 2, 3, 4, … } The Element Symbol m e a n s "is a n e le m e n t o f" m e a n s "is n o t a n e le m e n t o f" Examples: 3 2 , 3, 4 , 5 6 2 , 3, 4 , 5 Cardinal Number of a set We use the notation n(A) for the cardinality of a set A. Examples: If A={2, 4, 5, -3, 8, -7, 2.5}, the n(A)= 7, so A is finite X 1, 2, 3 , 1, 4, 5 , 3 Then n(x)= 3, X is finite If N is the set of all natural numbers, then N is infinite.