2 Chapter Introduction to Logic and Sets Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

2-1 Reasoning and Logic: An Introduction Students will be able to understand and explain • Quantifiers and their effects on statements. • Different forms of statements. • How to determine if two statements are logically equivalent. • How to develop logical arguments. • How to determine whether an argument is valid.

Definitions Statement – a sentence that is either true or false, but not both. Negation – a statement with the opposite truth value (logical value) of the given statement. The negation of a true statement is a false statement. If p is true, then ~ p is false.

Example Negate each of the following: 2 + 3 = 5 2 + 3 ≠ 5 A hexagon has six sides. A hexagon does not have six sides.

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.

Example 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.

Example continued Negate each of the following regardless of its truth value: c. There exists a natural number n such that 3n = 6. For all natural numbers, 3n ≠ 6. d. For all natural numbers n, 3n = 3n. There exists a natural number n such that 3n ≠ 3n.

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

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

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

Example 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.

Example (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.

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

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

Example 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.

A statement and its contrapositive are logically equivalent. p → q T F ~p ~ q ~ q → ~ p F T

Example Use truth tables to prove p → q ≅ ~q → ~p. p q ~p ~q p → q F p → q is equivalent to ~q → ~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 q → p (p → q) ∨ (q → p) T F

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.

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

Example 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.

Example (continued) We can use an Euler diagram to represent the validity of this reasoning. I

Example (continued) 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.

Example (continued) I Because people over 6 ft tall are outside the circle representing those who have influence and lobbyists are in the circle I, the conclusion is valid and no person over 6 ft tall is a lobbyist in Washington, D.C.

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.

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.

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.

Example Determine valid conclusions for the following true statements. a. If a triangle is equilateral, then it is isosceles. If a triangle is isosceles, it has at least two congruent sides. If a triangle is equilateral, then it has at least two congruent sides.

Example (continued) Determine valid conclusions for the following true statements. b. If a number is a whole number, then it is an integer. If a number is an integer, then it is a rational number. If a number is a rational number, then it is a real number. If a number is a whole number, then it is a real number.