Copyright © 2005 Pearson Education, Inc. Slide 1-1

Copyright © 2005 Pearson Education, Inc. Chapter 1

Copyright © 2005 Pearson Education, Inc. Slide 1-3 Quantitative Reasoning College Course Work Career Experiences Daily Life Quantitative Skills Prologue

Copyright © 2005 Pearson Education, Inc. Slide 1-4 Six Math Misconceptions 1. Math requires a special brain. 2. Math in modern issues is too complex. 3. Math makes you less sensitive. 4. Math makes no allowance for creativity. 5. Math provides exact answers. 6. Math is irrelevant to my life. Prologue

Copyright © 2005 Pearson Education, Inc. Slide 1-5 What is mathematics? Sum of its branchesWay to model the world Language Math Prologue

Copyright © 2005 Pearson Education, Inc. Slide 1-6 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 1-A

Copyright © 2005 Pearson Education, Inc. Slide 1-7 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. 1-A

Copyright © 2005 Pearson Education, Inc. Slide 1-8 Fallacy Structures Appeal to PopularityMany people believe p is true; therefore... p is true. False CauseA came before B; therefore... A caused B. Appeal to IgnoranceThere is no proof that p is true; therefore... p is false. Hasty GeneralizationA and B are linked one or a few times; therefore... A causes B or vice versa. Limited Choicep is false; therefore... only q can be true. 1-A

Copyright © 2005 Pearson Education, Inc. Slide 1-9 Fallacy Structures Appeal to Emotionp is associated with a positive emotional response; therefore... p is true. Personal AttackI have a problem with the person or group claiming p. p is not true. Circular Reasoningp 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 ManI 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. 1-A

Copyright © 2005 Pearson Education, Inc. Slide 1-10 Negations (Opposites) 1-B

Copyright © 2005 Pearson Education, Inc. Slide 1-11 And Statements (Conjunctions) Note: Conjunction is false unless both p and q are true. 1-B

Copyright © 2005 Pearson Education, Inc. Slide 1-12 Or Statements (Disjunctions) Note: Disjunction is true unless both p and q are false. 1-B

Copyright © 2005 Pearson Education, Inc. Slide 1-13 If... Then Statements (Conditionals) Note: Conditional is true unless p is true and q is false. 1-B

Copyright © 2005 Pearson Education, Inc. Slide 1-14 Truth Table Practice Note: ~ signifies NEGATION  signifies AND  signifies OR Practice by writing the truth values of each row in the table above. 1-B

Copyright © 2005 Pearson Education, Inc. Slide 1-15 Truth Table Practice Practice by writing the truth values of each row in the table above. 1-B Note: ~ signifies NEGATION  signifies AND  signifies OR

Copyright © 2005 Pearson Education, Inc. Slide 1-16 Truth Table Practice Practice by writing the truth values of each row in the table above. 1-B Note: ~ signifies NEGATION  signifies AND  signifies OR

Copyright © 2005 Pearson Education, Inc. Slide 1-17 Truth Table Practice Practice by writing the truth values of each row in the table above. 1-B Note: ~ signifies NEGATION  signifies AND  signifies OR

Copyright © 2005 Pearson Education, Inc. Slide 1-18 Truth Table Practice Practice by writing the truth values of each row in the table above. 1-B Note: ~ signifies NEGATION  signifies AND  signifies OR

Copyright © 2005 Pearson Education, Inc. Slide 1-19 Converse, Inverse, and Contrapositive 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. 1-B

Copyright © 2005 Pearson Education, Inc. Slide 1-20 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. 1-C

Copyright © 2005 Pearson Education, Inc. Slide 1-21 Real Number Venn Diagram 1-C

Copyright © 2005 Pearson Education, Inc. Slide 1-22 Venn Diagram for Categorical Propositions 1-C All S are P

Copyright © 2005 Pearson Education, Inc. Slide 1-23 Venn Diagram for Categorical Propositions 1-C No S are P

Copyright © 2005 Pearson Education, Inc. Slide 1-24 Venn Diagram for Categorical Propositions 1-C Some S are P

Copyright © 2005 Pearson Education, Inc. Slide 1-25 Venn Diagram for Categorical Propositions 1-C Some S are not P

Copyright © 2005 Pearson Education, Inc. Slide 1-26 Negations for Categorical Propositions 1-C PropositionNegation All S are P No S are P Some S are P Some S are not P Some S are P No S are P All S are P

Copyright © 2005 Pearson Education, Inc. Slide 1-27 Venn Diagram of Blood Types 1-C

Copyright © 2005 Pearson Education, Inc. Slide 1-28 Two Types of Arguments 1-D Inductive Reasoning specific premises general conclusion Deductive Reasoning general premises specific conclusion

Copyright © 2005 Pearson Education, Inc. Slide 1-29 Basic Forms of Conditional Deductive Arguments 1-D Valid (modus ponens) 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.

Copyright © 2005 Pearson Education, Inc. Slide 1-30 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 1-31 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 1-32 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 1-33 Inductive Counterexample 1-D 4 2  4  11  23 (prime)  5  11  31 (prime)  3  11  17 (prime)  2  11  13 (prime)  1  11  11 (prime)  0  11  11 (prime) 0 n 2  n  11 n Consider the following algebraic expression: n 2  n  11 It appears that n 2  n  11 will always equal a prime number when n ≥ 0. Or does it? How about n = 11? 11 2  = 121 (a non-prime counterexample)

Copyright © 2005 Pearson Education, Inc. Slide 1-34 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? 1-E