Exam1-Solutions

Exam1-Solutions - Math 326 - Exam 1 - 14 Sept 2010...

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Math 326 - Exam 1 - 14 Sept 2010 Solutions 1. Rewrite the following statements formally. (a) Every rational number can be written as the sum of two integers. ( x Q )( m,n Z )( x = m + n ) (b) Every valid argument with true premises has a true conclusion. ( argument x )( x has a true premise) = ( x has a true conclusion) (c) Every even integer greater than 2 can be written as the sum of two prime numbers. ( x Z )( x > 2 x is even)( p,q prime)( x = p + q ) 2. Negate each of the following: (a) For all integers n , if n is prime, then n is odd. ( n Z )( n is prime n is even) (b) x R , if x < 1 then 1 /x > 1. ( x R )(( x < 1) (1 /x 1)) (c) ( x A )( y B )( x = f ( y ) y = f ( x )). ( x A )( y B )( x 6 = f ( y ) y 6 = f ( x )) 3. (A) Which of the following is a negation for “Given any real numbers a and b , if a and b are rational, then a/b is rational.” (a) There exist real numbers a and b such that a and b are not rational and a/b is not rational. (b) Given any real numbers a and b ; if a and b are not rational then a/b is not rational. (c) There exist real numbers a and b such that a and b are not rational and a/b is rational. (d) Given any real numbers a and b ; if a and b are rational then a/b is not rational. (e) There exist real numbers a and b such that a and b are rational and a/b is not rational. (f) Given any real numbers a and b ; if a and b are not rational then a/b is rational. (B) Which of the following is the negation of the statement “For all real numbers r , there exists a number s such that rs > 10”: (a) For all real numbers r , there does not exist a number s such that rs > 10 (b) There exists real numbers
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This note was uploaded on 06/12/2011 for the course MATH 103 taught by Professor Wouters during the Spring '08 term at Wisc Oshkosh.

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Exam1-Solutions - Math 326 - Exam 1 - 14 Sept 2010...

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