Thinking Mathematically 13e

Thinking Mathematically 13e

Chapter 3 Introduction to Logic Copyright 2015, 2011, and 2007 Pearson Education, Inc. 1 Chapter 3: Introduction to Logic 3.1 3.2 3.3 3.4 3.5 3.6

Statements and Quantifiers Truth Tables and Equivalent Statements The Conditional and Circuits The Conditional and Related Statements Analyzing Arguments with Euler Diagrams Analyzing Arguments with Truth Tables Copyright 2016, 2012, and 2008 Pearson Education, Inc. 2 Section 3-1 Statements and Quantifiers Copyright 2016, 2012, and 2008 Pearson Education, Inc.

3 Statements and Qualifiers Distinguish between statements and nonstatements. Compose negations of statements. Translate between words and symbols. Interpret statements with quantifiers and form their negations. Find truth values of statements involving quantifiers and number sets. Copyright 2016, 2012, and 2008 Pearson Education, Inc. 4

Statements A statement is defined as a declarative sentence that is either true or false, but not both simultaneously. Copyright 2016, 2012, and 2008 Pearson Education, Inc. 5 Compound Statements A compound statement may be formed by combining two or more statements. The statements making up the compound statement are called the component statements. Various connectives, such as and, or, not, and ifthen, can be used in forming compound statements.

Copyright 2016, 2012, and 2008 Pearson Education, Inc. 6 Example: Deciding Whether a Statement is Compound Decide whether each statement is compound. a) You can pay me now, or you can pay me later. b) My pistol was made by Smith and Wesson. Solution a) The connective is or. The statement is compound. b) This is not compound since and is part of a manufacturer name and not a logical connective. Copyright 2016, 2012, and 2008 Pearson Education, Inc.

7 Negations The sentence Max has a valuable card is a statement; the negation of this statement is Max does not have a valuable card. The negation of a true statement is false and the negation of a false statement is true. Copyright 2016, 2012, and 2008 Pearson Education, Inc. 8 Example: Forming Negations Form the negation of each statement.

a. That city has a mayor. b. The moon is not a planet. Solution a. That city does not have a mayor. b. The moon is a planet. Copyright 2016, 2012, and 2008 Pearson Education, Inc. 9 Inequality Symbols Use the following inequality symbols for the next example. Symbolism

Meaning a b a is less than b a b a is greater than b a b a is less than or equal to b a b

a is greater than or equal to b Copyright 2016, 2012, and 2008 Pearson Education, Inc. 10 Example: Negating Inequalities Give a negation of each inequality. Do not use a slash symbol. a) x 9 b) 3 x 2 y 12 Solution a. x 9 b. 3 x 2 y 12

Copyright 2016, 2012, and 2008 Pearson Education, Inc. 11 Symbols To simplify work with logic, we use symbols. Statements are represented with letters, such as p, q, or r, while several symbols for connectives are shown below. Connective Symbol Type of Statement and

or not Conjunction Disjunction Negation ~ Copyright 2016, 2012, and 2008 Pearson Education, Inc. 12

Example: Translating from Symbols to Words Let p represent It is raining, and let q represent It is March. Write each symbolic statement in words. a) p q b) p q ( ) Solution a) It is raining or it is March. b) It is not the case that it is raining and it is

March. Copyright 2016, 2012, and 2008 Pearson Education, Inc. 13 Quantifiers The words all, each, every, and no(ne) are called universal quantifiers, while words and phrases such as some, there exists, and (for) at least one are called existential quantifiers. Quantifiers are used extensively in mathematics to indicate how many cases of a particular situation exist. Copyright 2016, 2012, and 2008 Pearson Education, Inc.

14 Negations of Quantified Statements Statement Negation All do. Some do not. Some do. None do. Copyright 2016, 2012, and 2008 Pearson Education, Inc.

15 Example: Forming Negations of Quantified Statements Form the negation of each statement. a) Some cats have fleas. b) Some cats do not have fleas. c) No cats have fleas. Solution a) No cats have fleas. b) All cats have fleas. c) Some cats have fleas. Copyright 2016, 2012, and 2008 Pearson Education, Inc.

16 Sets of Numbers p. 88 (chart) Natural (counting) {1, 2, 3, 4, } Whole numbers {0, 1, 2, 3, 4, } Integers {,3, 2, 1, 0, 1, 2, 3, } Rational numbers p p and q are integers and q 0 q

May be written as a terminating decimal, like 0.25, or a repeating decimal, like 0.333 Irrational {x | x is not expressible as a quotient of integers} Decimal representations never terminate and never repeat. Real numbers {x | x can be expressed as a decimal} Copyright 2016, 2012, and 2008 Pearson Education, Inc. 17 Example: Deciding Whether the Statements are True or False Decide whether each of the following statements about sets of numbers is true or false. a. There exists a whole number that is not a natural number.

b. Every integer is a natural number. Solution a. This is true (0 is it). b. This is false; 1 is an integer and not a natural number. Copyright 2016, 2012, and 2008 Pearson Education, Inc. 18

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