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Chapter 3 Introduction to Logic © 2008 Pearson Addison-Wesley. All rights reserved 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 More on the Conditional Analyzing Arguments with Euler Diagrams Analyzing Arguments with Truth Tables © 2008 Pearson Addison-Wesley. All rights reserved 3-3-2 Chapter 1 Section 3-3 The Conditional and Circuits © 2008 Pearson Addison-Wesley. All rights reserved 3-3-3 The Conditional and Circuits • Conditionals • Negation of a Conditional • Circuits © 2008 Pearson Addison-Wesley. All rights reserved 3-3-4 Conditionals A conditional statement is a compound statement that uses the connective if…then. The conditional is written with an arrow, so “if p then q” is symbolized p q. We read the above as “p implies q” or “if p then q.” The statement p is the antecedent, while q is the consequent. © 2008 Pearson Addison-Wesley. All rights reserved 3-3-5 Truth Table for The Conditional, If p, then q If p, then q pq p q T T T T F F F T T F F T © 2008 Pearson Addison-Wesley. All rights reserved 3-3-6 Special Characteristics of Conditional Statements 1. p q is false only when the antecedent is true and the consequent is false. 2. If the antecedent is false, then p q is automatically true. 3. If the consequent is true, then p q is automatically true. © 2008 Pearson Addison-Wesley. All rights reserved 3-3-7 Example: Determining Whether a Conditional Is True or False Decide whether each statement is True or False (T represents a true statement, F a false statement). a) T (4 2) b) (8 1) F Solution a) False b) True © 2008 Pearson Addison-Wesley. All rights reserved 3-3-8 Tautology A statement that is always true, no matter what the truth values of the components, is called a tautology. They may be checked by forming truth tables. © 2008 Pearson Addison-Wesley. All rights reserved 3-3-9 Negation of a Conditional p q The negation of p q is p © 2008 Pearson Addison-Wesley. All rights reserved q. 3-3-10 Writing a Conditional as an “or” Statement p q is equivalent to p q. © 2008 Pearson Addison-Wesley. All rights reserved 3-3-11 Example: Determining Negations Determine the negation of each statement. a) If you ask him, he will come. b) All dogs love bones. Solution a) You ask him and he will not come. b) It is a dog and it does not love bones. © 2008 Pearson Addison-Wesley. All rights reserved 3-3-12 Circuits Logic can be used to design electrical circuits. p p q Series circuit q Parallel circuit © 2008 Pearson Addison-Wesley. All rights reserved 3-3-13 Equivalent Statements Used to Simplify Circuits p q r p q p r p q r p q p r pq q p p q pq p p p p q p q p p p p q p q © 2008 Pearson Addison-Wesley. All rights reserved 3-3-14 Equivalent Statements Used to Simplify Circuits If T represents any true statement and F represents any false statement, then pT T pF F p p pT p F. © 2008 Pearson Addison-Wesley. All rights reserved 3-3-15 Example: Drawing a Circuit for a Conditional Statement Draw a circuit for p q r . Solution p q r p q r ~p q ~r © 2008 Pearson Addison-Wesley. All rights reserved 3-3-16