Copyright © 2005 Pearson Education, Inc. Slide 1-1 Chapter 1 Copyright © 2005 Pearson Education, Inc. Prologue Quantitative Reasoning Daily Life Quantitative Skills College Course Work Copyright © 2005 Pearson Education, Inc. Career Experiences Slide 1-3 Prologue Six Math Misconceptions 1. 2. 3. 4. 5. 6. Math requires a special brain. Math in modern issues is too complex. Math makes you less sensitive. Math makes no allowance for creativity. Math provides exact answers. Math is irrelevant to my life. Copyright © 2005 Pearson Education, Inc. Slide 1-4 Prologue What is mathematics? Language Math Sum of its branches Copyright © 2005 Pearson Education, Inc. Way to model the world Slide 1-5 1-A Importance of Logic Parade Magazine Ask Marilyn Column Question: “What is the most important thing a person can do to improve his or her critical thinking skills?” Phyllis Evitch, Rice Lake, Wisconsin Answer: “Study logic. Without a sound foundation in the principles of reasoning, you’ll be less able to understand your world, and the ramifications of this will ripple through everything from work to play. Even worse, you won’t realize what you’re missing.” Marilyn Vos Savant Copyright © 2005 Pearson Education, Inc. Slide 1-6 1-A Definitions Logic is the study of the methods and principles of reasoning. An argument uses a set of facts or assumptions, called premises, to support a conclusion. A fallacy is a deceptive argument—an argument in which the conclusion is not well supported by the premises. Copyright © 2005 Pearson Education, Inc. Slide 1-7 1-A Fallacy Structures Appeal to Popularity Many people believe p is true; therefore ... p is true. False Cause A came before B; therefore ... A caused B. Appeal to Ignorance There is no proof that p is true; therefore ... p is false. Hasty Generalization A and B are linked one or a few times; therefore ... A causes B or vice versa. Limited Choice p is false; therefore ... only q can be true. Copyright © 2005 Pearson Education, Inc. Slide 1-8 1-A Fallacy Structures Appeal to Emotion p is associated with a positive emotional response; therefore . . . p is true. Personal Attack I have a problem with the person or group claiming p. p is not true. Circular Reasoning p is true. p is restated in different words. Diversion (Red Herring) p is related to q and I have an argument concerning q; therefore . . . p is true. Straw Man I have an argument concerning a distorted version of p; therefore . . . I hope you are fooled into concluding I have an argument concerning the real version of p. Copyright © 2005 Pearson Education, Inc. Slide 1-9 1-B Negations (Opposites) p T F Copyright © 2005 Pearson Education, Inc. not p F T Slide 1-10 1-B And Statements (Conjunctions) p q p and q T T F F T F T F T F F F Note: Conjunction is false unless both p and q are true. Copyright © 2005 Pearson Education, Inc. Slide 1-11 1-B Or Statements (Disjunctions) p q p or q T T F F T F T F T T T F Note: Disjunction is true unless both p and q are false. Copyright © 2005 Pearson Education, Inc. Slide 1-12 If . . . Then Statements (Conditionals) 1-B p q if p, then q T T F F T F T F T F T T Note: Conditional is true unless p is true and q is false. Copyright © 2005 Pearson Education, Inc. Slide 1-13 1-B Truth Table Practice p q T T T F F T F F ~ p ~ q Note: ~ signifies NEGATION signifies AND signifies OR Copyright © 2005 Pearson Education, Inc. p q ~ p ~ q ~ ( p q) Practice by writing the truth values of each row in the table above. Slide 1-14 1-B Truth Table Practice p q ~ p ~ q p q T T F F T T F F T F F Note: ~ signifies NEGATION signifies AND signifies OR Copyright © 2005 Pearson Education, Inc. ~ p ~ q ~ ( p q) F F Practice by writing the truth values of each row in the table above. Slide 1-15 1-B Truth Table Practice p q ~ p ~ q p q T T F F T F F T F F T F T T F T F F Note: ~ signifies NEGATION signifies AND signifies OR Copyright © 2005 Pearson Education, Inc. ~ p ~ q ~ ( p q) Practice by writing the truth values of each row in the table above. Slide 1-16 1-B Truth Table Practice p q ~ p ~ q p q T T F F T F F T F F T F T T F T T F F T T F F Note: ~ signifies NEGATION signifies AND signifies OR Copyright © 2005 Pearson Education, Inc. ~ p ~ q ~ ( p q) Practice by writing the truth values of each row in the table above. Slide 1-17 1-B Truth Table Practice p q ~ p ~ q p q T T F F T F F T F F T F T T F T T F F T T F F T T F T T Note: ~ signifies NEGATION signifies AND signifies OR Copyright © 2005 Pearson Education, Inc. ~ p ~ q ~ ( p q) Practice by writing the truth values of each row in the table above. Slide 1-18 Converse, Inverse, and Contrapositive 1-B Conditional: If it is raining, then I will bring an umbrella to work. Converse: If I bring an umbrella to work, then it must be raining. Inverse: If it is not raining, then I will not bring an umbrella to work. Contrapositive: If I do not bring an umbrella to work, then it must not be raining. Copyright © 2005 Pearson Education, Inc. Slide 1-19 1-C Definitions A set is a collection of objects; the individual objects are the members of the set. We often describe sets by listing their members within a pair of braces, {}. If there are too many members to list, we can use three dots, …, to indicate a continuing pattern. Copyright © 2005 Pearson Education, Inc. Slide 1-20 1-C Real Number Venn Diagram Copyright © 2005 Pearson Education, Inc. Slide 1-21 Venn Diagram for Categorical Propositions 1-C All S are P Copyright © 2005 Pearson Education, Inc. Slide 1-22 Venn Diagram for Categorical Propositions 1-C No S are P Copyright © 2005 Pearson Education, Inc. Slide 1-23 Venn Diagram for Categorical Propositions 1-C Some S are P Copyright © 2005 Pearson Education, Inc. Slide 1-24 Venn Diagram for Categorical Propositions 1-C Some S are not P Copyright © 2005 Pearson Education, Inc. Slide 1-25 Negations for Categorical Propositions Proposition Negation All S are P Some S are not P No S are P Some S are P Some S are P No S are P Some S are not P All S are P Copyright © 2005 Pearson Education, Inc. 1-C Slide 1-26 1-C Venn Diagram of Blood Types Copyright © 2005 Pearson Education, Inc. Slide 1-27 1-D Two Types of Arguments Inductive Reasoning Deductive Reasoning specific premises general premises general conclusion specific conclusion Copyright © 2005 Pearson Education, Inc. Slide 1-28 Basic Forms of Conditional Deductive Arguments 1-D Affirming the Antecedent: If one gets a college degree, then one can get a good job. Marilyn has a college degree. Marilyn can get a good job. Valid (modus ponens) Copyright © 2005 Pearson Education, Inc. Slide 1-29 Basic Forms of Conditional Deductive Arguments 1-D Affirming the Consequent: If one gets a college degree, then one can get a good job. Marilyn gets a good job. Marilyn has a college degree. Invalid (inverse fallacy) Copyright © 2005 Pearson Education, Inc. Slide 1-30 Basic Forms of Conditional Deductive Arguments 1-D Denying the Antecedent: If one gets a college degree, then one can get a good job. Marilyn does not have a college degree. Marilyn cannot get a good job. Invalid (converse fallacy) Copyright © 2005 Pearson Education, Inc. Slide 1-31 Basic Forms of Conditional Deductive Arguments 1-D Denying the Consequent: If one gets a college degree, then one can get a good job. Marilyn does not have a good job. Marilyn does not a college degree. Valid (modus tollens) Copyright © 2005 Pearson Education, Inc. Slide 1-32 1-D Inductive Counterexample Consider the following algebraic expression: n2 n 11 It appears that n2 n 11 will always equal a prime number when n ≥ 0. n n2 n 11 0 02 0 11 11 (prime) 1 12 1 11 11 (prime) 2 22 2 11 13 (prime) Or does it? 3 32 3 11 17 (prime) How about n = 11? 4 42 4 11 23 (prime) 5 52 5 11 31 (prime) 112 11 + 11 = 121 (a non-prime counterexample) Copyright © 2005 Pearson Education, Inc. Slide 1-33 1-E Critical Thinking In Everyday Life General Guidelines. 1. Read (or listen) carefully. 2. Look for hidden assumptions. 3. Identify the real issue. 4. Use visual aids. 5. Understand all the options. 6. Watch for fine print and missing information. 7. Are other conclusions possible? Copyright © 2005 Pearson Education, Inc. Slide 1-34

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