1 Chapter An Introduction to Problem Solving Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

1-3 Reasoning and Logic: An Introduction Definitions Conditionals and Biconditionals Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definitions Statement – a sentence that is either true or false, but not both. Negation – a statement with the opposite truth value of the given statement. The negation of a true statement is a false statement. If p is true, then ~ p is false. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definitions Quantifier – words such as “all”, “some”, “every”, “there exists” Universal quantifier – applies to every element in a set. “All”, “every”, and “no” are universal quantifiers. Existential quantifier – applies to one or more (or possibly every) element in a set. “Some” and “there exists at least one” are existential quantifiers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-7 Negate each of the following: 2 + 3 = 5 2 + 3 ≠ 5 A hexagon has six sides. A hexagon does not have six sides. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-8 Negate each of the following: 1. All students like hamburgers. Some students do not like hamburgers. 2. Some people like mathematics. No people like mathematics. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-8, continued Negate each of the following: 3. There exists a natural number x such that 3x = 6. For all natural numbers, 3x ≠ 6. 4. For all natural numbers, 3x = 3x. There exists a natural number x such that 3x ≠ 3x. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Truth Tables A symbolic system to show all possible true-false patterns for statements. This is the truth table for negation. p ~p T F Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Conjunction A compound statement created from two given statements using the connective “and”. We use the symbol “” to represent “and”. A conjunction is true only if both statements are true; otherwise, it is false. p q p  q T F Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Disjunction A compound statement created from two given statements using the connective “or”. We use the symbol “” to represent “or”. A disjunction is false if both statements are false and true in all other cases. p q p  q T F Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-9 Classify each of the following as true or false: p: 2 + 3 = 5 q: 2 • 3 = 6 r: 5 + 3 = 9 p  q p is true and q is true, so p  q is true. 2. q  r q is true and r is false, so q  r is true. ~p  r ~p is false and r is false, so ~p  r is false. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-9 (continued) Classify each of the following as true or false: p: 2 + 3 = 5 q: 2 • 3 = 6 r: 5 + 3 = 9 4. ~p  ~q ~p is false and ~q is false, so ~p  ~q is false. 5. ~(p  q) p  q is true, so ~(p  q) is false. 6. (p  q)  ~r p  q is true and ~r is true, so (p  q)  ~r is true. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Conditionals and Biconditionals Conditional (or implication) – a statement expressed in the form “if p, then q.” Represented by p →q. p q p → q T F Hypothesis – the “if” part of the conditional Conclusion – the “then” part of the conditional Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Any implication p → q has three related implication statements: Statement if p, then q p → q Converse if q, then p q → p Inverse if not p, then not q ~p → ~ q Contrapositive if not q, then not p ~q → ~ p Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-11 Write the converse, inverse, and contrapositive for the following statement: If I am in San Francisco, then I am in California. Converse: If I am in California, then I am in San Francisco. Inverse: If I am not in San Francisco, then I am not in California. Contrapositive: If I am not in California, then I am not in San Francisco. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. A statement and its contrapositive are logically equivalent. p q p → q T F ~p ~ q ~ q → ~ p F T Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-12 Use truth tables to determine if p → q  ~q → ~p. p q ~p ~q p → q ~q → ~p T F p → q is equivalent to ~q → ~p. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Biconditional The conjunction of a statement and its converse. It is written as p ↔ q and is read “p if and only if q.” p q p → q q → p (p → q)  (q → p) T F Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Valid reasoning – if the conclusion follows unavoidably from true hypotheses. Example Hypotheses: All cats like fish. Felix is a cat. Conclusion: Therefore, Felix likes fish. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. We can use an Euler diagram to represent the validity of this reasoning. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-13 Determine if the following argument is valid: Hypotheses: In Washington, D.C., all lobbyists wear suits. No one in Washington, D.C., over 6 ft tall wears a suit. Conclusion: Persons over 6 ft tall are not lobbyists in Washington, D.C. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-13 (continued) We can use an Euler diagram to represent the validity of this reasoning. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-13 (continued) If L represents the lobbyists in Washington, D.C., and S the people who wear suits, the first hypothesis is pictured on the left. If W represents the people in Washington, D.C., over 6 ft tall, the second hypothesis is pictured on the right. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 1-13 (continued) Because people over 6 ft tall are outside the circle representing suit wearers and lobbyists are in the circle S, the conclusion is valid and no person over 6 ft tall is a lobbyist in Washington, D.C. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Law of Detachment (Modus Ponens) If the statement “if p, then q” is true, and p is true, then q is true. This is direct reasoning. Example Hypotheses: If it is raining, the grass is wet. It is raining. Conclusion: The grass is wet. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Indirect Reasoning (Modus Tollens) If a conditional is accepted as true, and the conclusion is false, then the hypothesis must be false. Example Hypotheses: If it is raining, the grass is wet. The grass is not wet. Conclusion: It is not raining. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Chain Rule If the statements “if p, then q” and “if q, then r” are true, then the statement “if p, then r” is true. Example Hypotheses: If you eat well, then you will be well. If you are well, then you are happy. Conclusion: If you eat well, then you are happy. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.