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Unformatted text preview: Chapter 2: The Logic of Compound Statements Section 2.1 9. ( n k ) ( n k ) 15. p q r q q r p ( q r ) T T T F T T T T F F F F T F T T T T T F F T T T F T T F T F F T F F F F F F T T T F F F F T T F The truth table shows that p ( q r ) and ( p q ) ( p r ) always have the same truth values. Therefore they are logically equivalent. This proves the distributive law for over . 24. p q r p q p r ( p q ) ( p r ) ( p q ) r T T T T T T T T T F T F T F T F T T T T T T F F T F T F F T T T F T T F T F T F T F F F T F F F F F F F F F F F  {z } different truth values The truth table shows that ( p q ) ( p r ) and ( p q ) r have different truth values in rows 2, 3, and 6. Hence they are not logically equivalent. 30. The dollar is not at an alltime high or the stock market is not at a record low. 33. 10 x or x 2 39. The statements logical form is ( p q ) (( r s ) t ) , so its negation has the form (( p q ) (( r s ) t )) ( p q ) (( r s ) t )) ( p q ) ( ( r s ) t )) ( p q ) (( r s ) t )) . Thus a negation is ( num orders 50 or num instock 300) and ((50 &gt; num orders or num orders 75) or num instock 500). 4 Solutions for Exercises: The Logic of Compound Statements 42. p q r p q p q q r (( p q ) ( q r )) (( p q ) ( q r )) q T T T F F F T F F T T F F F F F F F T F T F T F F F F T F F F T F F F F F T T T F T T T F F T F T F T F F F F F T T T F F F F F F F T T F F F F  {z } all F s Since all the truth values of (( p q ) ( q r )) q are F , (( p q ) ( q r )) q is a contradiction. 45. Let b be Bob is a double math and computer science major, m be Ann is a math major, and a be Ann is a double math and computer science major. Then the two statements can be symbolized as follows: a . ( b m ) a and b . ( b a ) ( m b ) . Note : The entries in the truth table assume that a person who is a double math and computer science major is also a math major and a computer science major. b m a a b m m b b a ( b a ) ( b m ) a ( b a ) ( m b ) T T T F T T T F F F T T F T F T F T T T T F T F T F T F F F T F F T F F F T F F F T T F F F F T F F F T F T F F F T F F F F T F F F F T F F F F F T F F F T F F  {z } same truth values The truth table shows that ( b m ) a and ( b a ) ( m b ) always have the same truth values. Hence they are logically equivalent....
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 Spring '11
 Shirley
 Logic

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