1. Chapter 2 Section 5 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND

2. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 5 - Slide 2 Chapter 2 Sets

3. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 5 - Slide 3 WHAT YOU WILL LEARN Application of sets

4. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 5 - Slide 4 Section 5 Applications of Sets

5. Chapter 2 Section 5 - Slide 5 Copyright © 2009 Pearson Education, Inc. Example: Toothpaste Taste Test A drug company is considering manufacturing a new toothpaste. They are considering two flavors, regular and mint. In a sample of 120 people, it was found that 74 liked the regular, 62 liked the mint, and 35 liked both types. How many liked only the regular flavor? How many liked either one or the other or both? How many people did not like either flavor?

6. Chapter 2 Section 5 - Slide 6 Copyright © 2009 Pearson Education, Inc. Solution Begin by setting up a Venn diagram with sets A (regular flavor) and B (mint flavor). Since some people liked both flavors, the sets will overlap and the number who liked both with be placed in region II. 35 people liked both flavors. U A(Regular) B(Mint) 35 II

7. Chapter 2 Section 5 - Slide 7 Copyright © 2009 Pearson Education, Inc. Solution (continued) Next, region I will refer to those who liked only the regular and region III will refer to those who liked only the mint. In order to get the number of people in each region, find the difference between all the people who liked each toothpaste and those who liked both. I: 74 – 35 = 39 III: 62 – 35 = 27 U A(Regular) B(Mint) 39 regular only 27 mint only III II I both 35

8. Chapter 2 Section 5 - Slide 8 Copyright © 2009 Pearson Education, Inc. Solution (continued) “One or the other or both” represents the UNION of the two sets. Therefore, 39 + 27 + 35 = 101 people who liked one or the other or both.

9. Chapter 2 Section 5 - Slide 9 Copyright © 2009 Pearson Education, Inc. Solution (continued) Take the total number of people in the entire sample and subtract the number who liked one or the other or both. 120-101 = 19 people did not like either flavor. U A(Regular) B(Mint) 35 both 62-35=27 Likedmint only 74-35=39 Likedregular only 19 liked neither I I I I I

10. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 5 - Slide 10 Chapter 3 Logic

11. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 5 - Slide 11 WHAT YOU WILL LEARN Statements, quantifiers, and compound statements Statements involving the words not, and, or, if… then…, and if and only if

12. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 5 - Slide 12 Section 1 Statements and Logical Connectives

13. Chapter 2 Section 5 - Slide 13 Copyright © 2009 Pearson Education, Inc. HISTORY—The Greeks: Aristotelian logic: The ancient Greeks were the first people to look at the way humans think and draw conclusions. Aristotle (384-322 B.C.) is called the father of logic. This logic has been taught and studied for more than 2000 years.

14. Chapter 2 Section 5 - Slide 14 Copyright © 2009 Pearson Education, Inc. Mathematicians Gottfried Wilhelm Leibniz (1646-1716) believed that all mathematical and scientific concepts could be derived from logic. He was the first to seriously study symbolic logic. In this type of logic, written statements use symbols and letters. George Boole (1815 – 1864) is said to be the founder of symbolic logic because he had such impressive work in this area.

15. Chapter 2 Section 5 - Slide 15 Copyright © 2009 Pearson Education, Inc. Logic and the English Language Connectives - words such as and, or, if, then Exclusive or - one or the other of the given events can happen, but not both. Inclusive or - one or the other or both of the given events can happen.

16. Chapter 2 Section 5 - Slide 16 Copyright © 2009 Pearson Education, Inc. Statements and Logical Connectives Statement - A sentence that can be judged either true or false.  Labeling a statement true or false is called assigning a truth value to the statement. Simple Statements - A sentence that conveys only one idea and can be assigned a truth value. Compound Statements - Sentences that combine two or more simple statements.

17. Chapter 2 Section 5 - Slide 17 Copyright © 2009 Pearson Education, Inc. Negation of a Statement Negation of a statement – change a statement to its opposite meaning.  The negation of a false statement is always a true statement.  The negation of a true statement is always a false statement.

18. Chapter 2 Section 5 - Slide 18 Copyright © 2009 Pearson Education, Inc. Quantifiers Quantifiers - words such as all, none, no, some, etc… Be careful when negating statements that contain quantifiers.

19. Chapter 2 Section 5 - Slide 19 Copyright © 2009 Pearson Education, Inc. Negation of Quantified Statements Form of statement All are. None are. Some are. Some are not. Form of negation Some are not. Some are. None are. All are. None are. Some are not. All are. Some are.

20. Chapter 2 Section 5 - Slide 20 Copyright © 2009 Pearson Education, Inc. Example: Write Negations Write the negation of the statement. Some candy bars contain nuts. Solution:Since some means “at least one” this statement is true. The negation is “No candy bars contain nuts,” which is a false statement.

21. Chapter 2 Section 5 - Slide 21 Copyright © 2009 Pearson Education, Inc. Example: Write Negations continued Write the negation of the statement. All tables are oval. Solution:This is a false statement since some tables are round, rectangular, or other shapes. The negation would be “Some tables are not oval.”

22. Chapter 2 Section 5 - Slide 22 Copyright © 2009 Pearson Education, Inc. Compound Statements Statements consisting of two or more simple statements are called compound statements. The connectives often used to join two simple statements are and, or, if…then…, and if and only if.

23. Chapter 2 Section 5 - Slide 23 Copyright © 2009 Pearson Education, Inc. Not Statements (Negation) The symbol used in logic to show the negation of a statement is ~. It is read “not”.  Example: The negation of p is: ~ p.

24. Chapter 2 Section 5 - Slide 24 Copyright © 2009 Pearson Education, Inc. And Statements (Conjunction) is the symbol for a conjunction and is read “and.” The other words that may be used to express a conjunction are: but, however, and nevertheless.  Example: The conjunction of p and q is: p ^ q.

25. Chapter 2 Section 5 - Slide 25 Copyright © 2009 Pearson Education, Inc. Example: Write a Conjunction Write the following conjunction in symbolic form: The dog is gray, but the dog is not old. Solution: Let p and q represent the simple statements. p: The dog is gray. q: The dog is old. In symbolic form, the compound statement is p Λ ~ q

26. Chapter 2 Section 5 - Slide 26 Copyright © 2009 Pearson Education, Inc. Or Statements (Disjunction) The disjunction is symbolized by and read “or.” In this book the “or” will be the inclusive or (except where indicated in the exercise set).  Example: The disjunction of p and q is: p V q.

27. Chapter 2 Section 5 - Slide 27 Copyright © 2009 Pearson Education, Inc. Example: Write a Disjunction Write the statement in symbolic form. Carl will not go to the movies or Carl will not go to the baseball game. Solution: Let p and q represent the simple statements. p: Carl will go to the movies. q: Carl will go to the baseball game. In symbolic form, the compound statement is

28. Chapter 2 Section 5 - Slide 28 Copyright © 2009 Pearson Education, Inc. If-Then Statements (continued) The conditional is symbolized by and is read “if-then.”  The antecedent is the part of the statement that comes before the arrow.  The consequent is the part that follows the arrow.  Example: If p, then q is symbolized as: p  q.

29. Chapter 2 Section 5 - Slide 29 Copyright © 2009 Pearson Education, Inc. Example: Write a Conditional Statement Let p: Nathan goes to the park. q: Nathan will swing. Write the following statements symbolically. a.If Nathan goes to the park, then he will swing. b.If Nathan does not go to the park, then he will not swing. Solutions a) b)

30. Chapter 2 Section 5 - Slide 30 Copyright © 2009 Pearson Education, Inc. If and Only If Statements (Biconditional) The biconditional is symbolized by and is read “if and only if.” If and only if is sometimes abbreviated as “iff.” The biconditional p  q is the conjunction of the two conditionals p  q and q  p: p  q = (p  q) (q  p)

31. Chapter 2 Section 5 - Slide 31 Copyright © 2009 Pearson Education, Inc. Example: Write a Statement Using the Biconditional Let p: The dryer is running. q: There are clothes in the dryer. Write the following symbolic statements in words. a) b) Solutions: a.There are clothes in the dryer if and only if the dryer is running. b.It is false that the dryer is running if and only if there are no clothes in the dryer.