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Set Theory

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.

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Set Theory

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  1. Set Theory Using Mathematics to Classify Objects

  2. 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

  3. A setis 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 methodwhich 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”

  4. 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

  5. Set-builder notation:

  6. 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}

  7. 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}

  8. 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

  9. Universal Set • Example: Consider female consumers living in the U.S. • The universal set is • Example: Consider the set of Natural numbers. • The universal set is • U = { 1, 2, 3, 4, … }

  10. The Element Symbol • Examples:

  11. Cardinal Numberof 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 Then n(x)= 3, X is finite If N is the set of all natural numbers, then N is infinite.

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