SG4eCh2 - Chapter 2 The Logic of Compound Statements...

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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 all-time 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 &amp;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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SG4eCh2 - Chapter 2 The Logic of Compound Statements...

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