Set Theory

1. Set Theory Using Mathematics to Classify Objects

2. The Language of Sets 2.1 • Specify sets using both listing and set-builder notation • Understand when sets are well-defined • Use the element symbol property (continued on next slide)

3. The Language of Sets 2.1 • Find the cardinal number of sets

4. Representing Sets • Set – collection of objects • Element – a member of a set

5. Representing Sets • Set-builder notation:

6. Use an alternative method to write each set. • T = {A-, A+, B-, B+, AB-, AB+, O-, O+} • B= {y:y is a color of the American flag} • A = {a:a is a counting number less than 20 and is evenly divisible by 3}

7. Sets of Numbers Commonly Used in Mathematics • N = {x:x is a natural number} = {1, 2, 3, . . .} • W = {x:x is a whole number} = {0, 1, 2, 3, . . .} • I = {x:x is an integer} = {. . . , -2, -1, 0, 1, 2, . . .} • Q = {x:x is a rational number} = {x:x is of the form where a and b are integers and b ≠ 0} • R = {x:x is a real number} = {x:x has a decimal expansion}

8. Representing Sets • A set is well-defined if we are able to tell whether any particular object is an element of the set. • Example: Which sets are well-defined? • (a) • (b)

9. Quiz Yourself • Which sets are well-defined? • {x:x is a mountain over 10,000 feet high} • {y:y is a large number}

10. Representing Sets • 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

11. Representing Sets • Example: Consider female consumers living in the U.S. The universal set is

12. The Element Symbol • Example:

13. Which symbol belongs in the following? • 3 ? {2, 3, 4, 5} • {5} ? {2, 3, 4, 5} • Bill Gates ? {x:x is a billionaire} • Jogging ? {y:y is an aerobic exercise} • The ace of hearts ? {f:f is a face card}

14. Quiz Yourself • True or False? • 3 {x:x is an odd counting number} • 2 • Skim milk {x:x is a food containing calcium}

15. Cardinal Number • Example: State the cardinal number of the set.

16. State whether each set is finite or infinite. If it is finite, give its cardinal number • P = {x:x is a planet in our solar system} • N = {1, 2, 3, . . . } • A = {y:y is a person living in the United States who is not a citizen}

17. Quiz Yourself • Find the cardinal number of each set. • {2, 4, . . ., 20} • {{1, 2}, {1, 3, 4}} • {s:s is one of the United States}