hw5solsmc - is odd) ( b is odd) . In words this is just a...

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Solutions to Homework 5 - Math 2000 The solutions to 2.39,2.43,2.45,2.49,,2.53,2.55 may be found in the book. Exercises 2.46,2.48 (a),(c) were done during the tutorial. (# 2.40) Write negations of the following open sentences. (a) Either x = 0 or y = 0. (b) The integers a and b are both even. Solution . (a) Let P ( x ) , Q ( y ) be the open sentences P ( x ) : x = 0 , Q ( y ) : y = 0 . The first open sentence is just P ( x ) Q ( y ) . It’s negation is ( P ( x ) Q ( y )) ≡∼ P ( x ) ∧ ∼ Q ( y ) ( x 6 = 0) ( y 6 = 0) . In words this is just ‘x is not equal to zero and y is not equal to zero.’ or ‘x and y are not equal to zero.’ (b) Let P ( a ) , Q ( b ) be the open sentences P ( a ) : a is even , Q ( b ) : b is even . The first open sentence is just P ( a ) Q ( b ) . It’s negation is ( P ( a ) Q ( b )) ≡∼ P ( a ) ∨ ∼ Q ( b ) ( a
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Unformatted text preview: is odd) ( b is odd) . In words this is just a is odd or b is odd. or a or b is odd. (# 2.48) Determine the truth value of each of the following statements. (b) n N , n + 1 2. (d) x R , 3 x 2-27. Solution . (b) This is a true statement. Note that N is just the set of positive integers { 1 , 2 , 3 , . . . } . Thus if n N then n 1 is true. Thus adding 1 to each side of the inequality n + 1 1 + 1 = 2 is true for all n N . (d) This is a true statement. Note that if x = 3 then 3(3) 2 = 3 9 = 27. Thus there exists a real number which satises 3 x 2 = 27. 1...
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