2 1 Reasoning and Logic An Introduction Students

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2 -1 Reasoning and Logic: An Introduction Students will be able to understand explain

2 -1 Reasoning and Logic: An Introduction Students will be able to understand explain • Quantifiers and their effects on statements. • Different forms of statements. • How to determine whether two statements are logically equivalent. • How to develop logical arguments. • How to determine whether an argument is valid. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 1

Definitions Statement – a sentence that is either true or false, but not both.

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. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 2

Example 1 Negate each of the following statements: 1. 2 + 3 = 5

Example 1 Negate each of the following statements: 1. 2 + 3 = 5 2+3≠ 5 2. A hexagon has six sides. A hexagon does not have six sides. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 3

Definitions Quantifier – words such as “all”, “some”, “every”, “there exists”, and “no” (or

Definitions Quantifier – words such as “all”, “some”, “every”, “there exists”, and “no” (or none). 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. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 4

Example 2 Negate each of the following regardless of its truth value: a. All

Example 2 Negate each of the following regardless of its truth value: a. All students like hamburgers. Some students do not like hamburgers. b. Some people like mathematics. No people like mathematics. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 5

Example 2 continued Negate each of the following regardless of its truth value: c.

Example 2 continued Negate each of the following regardless of its truth value: c. There exists a natural number n such that 3 n = 6. For all natural numbers n, 3 n ≠ 6. d. For all natural numbers n, 3 n = 3 n. There exists a natural number n such that 3 n ≠ 3 n. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 6

Truth Tables Update table. A symbolic system to show all possible truefalse patterns for

Truth Tables Update table. A symbolic system to show all possible truefalse patterns for statements. This is the truth table for negation. Statement p T F ALWAYS LEARNING Negation ~p F T Copyright © 2020, 2016, 2013 Pearson Education, Inc. 7

Conjunction A compound statement created from two given statements using the connective “and”. We

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 T T F F ALWAYS LEARNING q T F p q T F F F Copyright © 2020, 2016, 2013 Pearson Education, Inc. 8

Disjunction A compound statement created from two given statements using the connective “or”. We

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 T T F F q T F ALWAYS LEARNING p q T T T F Copyright © 2020, 2016, 2013 Pearson Education, Inc. 9

Example 3 Classify each of the following as true or false: p: 2 +

Example 3 Classify each of the following as true or false: p: 2 + 3 = 5 q: 2 • 3 = 6 r: 5 + 3 = 9 a. p q p is true and q is true, so p q is true. b. q r q is true and r is false, so q r is true. c. ~p r ~p is false and r is false, so ~p r is false. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 10

Example 3 (continued) Classify each of the following as true or false: p: 2

Example 3 (continued) Classify each of the following as true or false: p: 2 + 3 = 5 q: 2 • 3 = 6 r: 5 + 3 = 9 d. ~p ~q ~p is false and ~q is false, so ~p ~q is false. e. ~(p q) p q is true, so ~(p q) is false. f. (p q) ~r p q is true and ~r is true, so (p q) ~r is true. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 11

Conditionals and Biconditionals Conditional (or implication) – a statement expressed in the form “if

Conditionals and Biconditionals Conditional (or implication) – a statement expressed in the form “if p, then q. ” Represented by p →q. p T T F F q T F p→q T F T T Hypothesis – the “if” part of the conditional Conclusion – the “then” part of the conditional ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 12

Any implication p → q has three related implication statements: Statement If p, then

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 ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 13

Example 6 Write the converse, inverse, and contrapositive for the following statement: If I

Example 6 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. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 14

Example 7 Update table. Use truth tables to prove p → q ~q →

Example 7 Update table. Use truth tables to prove p → q ~q → ~p. p q p→q ~q ~p ~q → ~p T T F F T F T F F T T T F T T p → q is equivalent to ~q → ~p. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 15

Biconditional The conjunction of a statement and its converse. It is written as p

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 T T F F T F T T ALWAYS LEARNING q → p (p → q) (q → p) T T F F T Copyright © 2020, 2016, 2013 Pearson Education, Inc. 16

Valid reasoning – if the conclusion follows unavoidably from true hypotheses. Example Hypotheses: Conclusion:

Valid reasoning – if the conclusion follows unavoidably from true hypotheses. Example Hypotheses: Conclusion: ALWAYS LEARNING All cats like fish. Felix is a cat. Therefore, Felix likes fish. Copyright © 2020, 2016, 2013 Pearson Education, Inc. 17

We can use an Euler diagram to represent the validity of this reasoning. ALWAYS

We can use an Euler diagram to represent the validity of this reasoning. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 18

Example 8 Determine if the following argument is valid: Hypotheses: In Washington, D. C.

Example 8 Determine if the following argument is valid: Hypotheses: In Washington, D. C. , all lobbyists have influence. No one in Washington, D. C. , over 6 ft tall has influence. Conclusion: Persons over 6 ft tall are not lobbyists in Washington, D. C. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 19

Example 8 (continued) We can use an Euler diagram to represent the validity of

Example 8 (continued) We can use an Euler diagram to represent the validity of this reasoning. I ALWAYS LEARNING I Copyright © 2020, 2016, 2013 Pearson Education, Inc. 20

Example 8 (continued) I I If L represents the lobbyists in Washington, D. C.

Example 8 (continued) I I If L represents the lobbyists in Washington, D. C. , and I the people who have influence, 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. ALWAYS LEARNING Copyright © 2020, 2016, 2013 Pearson Education, Inc. 21