ExtraExamples_1_5

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Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples Section 1.5—Rules of Inference Page references correspond to locations of Extra Examples icons in the textbook. p.67, icon at Example 6 #1. The proposition ( ¬ q ( p q )) → ¬ p is a tautology, as the reader can check. It is the basis for the rule of inference modus tollens : ¬ q p q . . . ¬ p Suppose we are given the propositions: “If the class finishes Chapter 2, then they have a quiz” and “The class does not have a quiz.” Find a conclusion that can be drawn using modus tollens. Solution: Let p represent “The class finishes Chapter 2” and q represent “The class has a quiz.” According to modus tollens, because we have ¬ q and p q , we can conclude ¬ p , or “The class did not finish Chapter 2.” p.67, icon at Example 6 #2. Suppose that “it is snowing” is true and that “it is windy” is true. Using the conjunction rule, what conclusion can be drawn? Solution: Using s for “it is snowing” and w for “it is windy,” we are given that s is true and w is true. By the conjunction rule, we can conclude s w , or “it is snowing and windy”. p.67, icon at Example 6 #3. Suppose “I have a dime or a quarter in my pocket” and “I do not have a dime in my pocket.” According to the disjunctive syllogism rule, what can we conclude? Solution: Using d for “I have a dime in my pocket” and q for “I have a quarter in my pocket”, we are given d q and ¬ d . According to the disjunctive syllogism rule, we can conclude

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