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3 Logic The Study of What’s True or False or Somewhere in Between Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 1 3.2 Truth Tables • Learn the truth tables for the five fundamental connectives • Compute truth tables for compound statements (continued on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 2 3.2 Truth Tables • Determine when statements are logically equivalent • State and apply DeMorgan’s laws. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 3 Truth Tables • We want to know if a pair of similar statements such as p q and p q mean the same thing. • We use truth tables to determine when compound statements are true and when they are false, and whether a pair of statements have the same meaning. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 4 Truth Tables Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 5 Truth Tables • If p is true, then ~p is false. • If p is false, then ~p is true. (example on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 6 Truth Tables • Example: Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 7 Truth Tables • A conjunction is true only when both of its components are true. (example on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 8 Truth Tables • Example: (solution on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 9 Truth Tables • Example: (solution on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 10 Truth Tables • Solution: Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 11 Truth Tables • A disjunction is false only when both p and q are false. (example on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 12 Truth Tables • Example: (solution on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 13 Truth Tables • Solution: Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 14 Truth Tables • Because of their similar structures, there are parallels that occur in logic and set theory. Often if a result is true for set theory, then a similar result also holds for logic (and vice versa). Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 15 Compound Statements • We use truth tables to find the logical values of complex statements. • Example: Compute a truth table for p q p q . (solution on next 3 slides) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 16 Compound Statements p q p q . (solution continued on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 17 Compound Statements p q p q . (solution continued on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 18 Compound Statements p q p q . If p is false and q is true, line 3 tells us that is false. Copyright © 2014, 2010, 2007 Pearson Education, Inc. p q p q Section 3.2, Slide 19 Compound Statements • Definition: If the final column of a truth table contains all T’s, then the statement is always true. Such a statement is called a tautology. • Example: “This contestant will win or will not win.” p T p F F T p p T T Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 20 Compound Statements How many lines should your truth table have? Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 21 Logically Equivalent Statements • Logically equivalent statements express the same meaning. (example on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 22 Logically Equivalent Statements • Example: a) pd b) p d (solution on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 23 Logically Equivalent Statements • Solution: a) pd b) p d The two statements are logically equivalent. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 24 DeMorgan’s Laws (example on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 25 DeMorgan’s Laws • Example: Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 26 DeMorgan’s Laws Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 27 Truth Tables: Alternative Method For p q p q : Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.2, Slide 28