Chapter 3 Section 4 – Slide 1 Copyright © 2009 Pearson Education, Inc. AND

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 2 Chapter 3 Logic

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 3 WHAT YOU WILL LEARN Statements, quantifiers, and compound statements Statements involving the words not, and, or, if… then…, and if and only if Truth tables for negations, conjunctions, disjunctions, conditional statements, and biconditional statements Self-contradictions, tautologies, and implications

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 4 WHAT YOU WILL LEARN Equivalent statements, De Morgan’s law, and variations of conditional statements Symbolic arguments and standard forms of arguments Euler diagrams and syllogistic arguments Using logic to analyze switching circuits

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 5 Section 4 Equivalent Statements

Chapter 3 Section 4 – Slide 6 Copyright © 2009 Pearson Education, Inc. Equivalent Statements Two statements are equivalent if both statements have exactly the same truth values in the answer columns of the truth tables.  In a truth table, if the answer columns are identical, the statements are equivalent. If the answer columns are not identical, the statements are not equivalent. Sometimes the words logically equivalent are used in place of the word equivalent.

Chapter 3 Section 4 – Slide 7 Copyright © 2009 Pearson Education, Inc. De Morgan’s Laws

Chapter 3 Section 4 – Slide 8 Copyright © 2009 Pearson Education, Inc. Example: Using De Morgan’s Laws to Write an Equivalent Statement Use De Morgan’s laws to write a statement logically equivalent to “Benjamin Franklin was not a U.S. president, but he signed the Declaration of Independence.” Solution: Let p: Benjamin Franklin was a U.S. president The statement symbolically is ~p V q. q: Benjamin Franklin signed the Declaration of Independence

Chapter 3 Section 4 – Slide 9 Copyright © 2009 Pearson Education, Inc. Example: Using De Morgan’s Laws to Write an Equivalent Statement (continued) Therefore, the logically equivalent statement to the given statement is: “It is false that Benjamin Franklin was a U.S. president or Benjamin Franklin did not sign the Declaration of Independence.” The logically equivalent statement in symbolic form is

Chapter 3 Section 4 – Slide 10 Copyright © 2009 Pearson Education, Inc. To change a conditional statement into a disjunction, negate the antecedent, change the conditional symbol to a disjunction symbol, and keep the consequent the same. To change a disjunction statement to a conditional statement, negate the first statement, change the disjunction symbol to a conditional symbol, and keep the second statement the same. Switching Between a Conditional and a Disjunction

Chapter 3 Section 4 – Slide 11 Copyright © 2009 Pearson Education, Inc. Variations of the Conditional Statement The variations of conditional statements are the converse of the conditional, the inverse of the conditional, and the contrapositive of the conditional.

Chapter 3 Section 4 – Slide 12 Copyright © 2009 Pearson Education, Inc. “if not q, then not p”~p~q Contrapositive of the conditional “if not p, then not q” ~q~p Inverse of the conditional “if q, then p” pq Converse of the conditional “if p, then q”qpConditional Read Symbolic Form Name Variations of the Conditional Statement

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 13 Section 5 Symbolic Arguments

Chapter 3 Section 4 – Slide 14 Copyright © 2009 Pearson Education, Inc. Symbolic Arguments An argument is valid when its conclusion necessarily follows from a given set of premises. An argument is invalid (or a fallacy) when the conclusion does not necessarily follow from the given set of premises.

Chapter 3 Section 4 – Slide 15 Copyright © 2009 Pearson Education, Inc. Law of Detachment Also called modus ponens. The argument form symbolically written: Premise 1: Premise 2: If [ premise 1 and premise 2 ] then conclusion Conclusion: [ (p  q)  p ]  q

Chapter 3 Section 4 – Slide 16 Copyright © 2009 Pearson Education, Inc. Determine Whether an Argument is Valid Write the argument in symbolic form. Compare the form with forms that are known to be either valid or invalid. If the argument contains two premises, write a conditional statement of the form [(premise 1)  (premise 2)]  conclusion Construct a truth table for the statement above. If the answer column of the table has all trues, the statement is a tautology, and the argument is valid. If the answer column of the table does not have all trues, the argument is invalid.

Chapter 3 Section 4 – Slide 17 Copyright © 2009 Pearson Education, Inc. Example: Determining Validity with a Truth Table Determine whether the following argument is valid or invalid. If you score 90% on the final exam, then you will get an A for the course. You will not get an A for the course.   You do not score 90% on the final exam.

Chapter 3 Section 4 – Slide 18 Copyright © 2009 Pearson Education, Inc. Example: Determining Validity with a Truth Table (continued) Construct a truth table. In symbolic form the argument is: Solution: Let p: You score 90% on the final exam. q: You will get an A in the course. p  q~q~pp  q~q~p

Chapter 3 Section 4 – Slide 19 Copyright © 2009 Pearson Education, Inc. Example: Determining Validity with a Truth Table (continued) Fill-in the table in order, as follows: Since column 7 has all T’s, the argument is valid. pq[(p  q)q)  ~q]  ~p~p TTTTTFFTF TFTFFFTTF FTFTTFFTT FFFTFTTTT

Chapter 3 Section 4 – Slide 20 Copyright © 2009 Pearson Education, Inc. Valid Arguments Law of Detachment Law of Syllogism Law of Contraposition Disjunctive Syllogism

Chapter 3 Section 4 – Slide 21 Copyright © 2009 Pearson Education, Inc. Invalid Arguments Fallacy of the Converse Fallacy of the Inverse