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

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

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 3 - Slide 3 WHAT YOU WILL LEARN Truth tables for conditional statements, and biconditional statements Self-contradictions, tautologies, and implications Equivalent statements, De Morgan’s law, and variations of conditional statements

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 3 - Slide 4 Section 3 Truth Tables for the Conditional and Biconditional

Chapter 3 Section 3 - Slide 5 Copyright © 2009 Pearson Education, Inc. Conditional The conditional statement p  q is true in every case except when p is a true statement and q is a false statement. TFFCase 4 TTFCase 3 FFTCase 2 TTTCase 1 qp

Chapter 3 Section 3 - Slide 6 Copyright © 2009 Pearson Education, Inc. Biconditional The biconditional statement, p↔q means that p  q and q  p or, symbolically (p  q)  (q  p) order of steps FTFTFTFFFcase 4 FFTFTTFTFcase 3 TTFFFFTFTcase 2 TTTTTTTTTcase 1 p)(qq)(pqp

Chapter 3 Section 3 - Slide 7 Copyright © 2009 Pearson Education, Inc. Example: Truth Table with a Conditional Construct a truth table for ~p  ~q. Solution: Construct standard four case truth table. pq~p~p  ~q~q TTFFTTFF TFTFTFTF F F T T T T F T F T F T Then fill-in the table in order, as follows: It’s a conditional, the answer lies under the 

Chapter 3 Section 3 - Slide 8 Copyright © 2009 Pearson Education, Inc. Self-Contradiction A self-contradiction is a compound statement that is always false.  When every truth value in the answer column of the truth table is false, then the statement is a self-contradiction.

Chapter 3 Section 3 - Slide 9 Copyright © 2009 Pearson Education, Inc. Tautology A tautology is a compound statement that is always true.  When every truth value in the answer column of the truth table is true, the statement is a tautology.

Chapter 3 Section 3 - Slide 10 Copyright © 2009 Pearson Education, Inc. Implication An implication is a conditional statement that is a tautology.  The consequent will be true whenever the antecedent is true.

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 3 - Slide 11 Section 4 Equivalent Statements

Chapter 3 Section 3 - Slide 12 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 3 - Slide 13 Copyright © 2009 Pearson Education, Inc. De Morgan’s Laws

Chapter 3 Section 3 - Slide 14 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  q. q: Benjamin Franklin signed the Declaration of Independence

Chapter 3 Section 3 - Slide 15 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 3 - Slide 16 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 3 - Slide 17 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 3 - Slide 18 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