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hw4solsmc - Solutions to Homework 4 Math 2000 2.18 Consider...

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Solutions to Homework 4 - Math 2000 (# 2.18) Consider the open sentences P ( n ) : 5 n + 3 is prime. and Q ( n ) : 7 n + 1 is prime over the domain N . (a). State P ( n ) = Q ( n ) in words. (b). State P (2) = Q (2) in words. Is this statement true or false? (c). State P (6) = Q (6) in words. Is this statement true or false? Solution . (a). If 5 n + 1 is prime, then 7 n + 1 is prime. (b). If 5(2) + 1 is prime, then 7(2) + 1 is prime. Since 5(2) + 1 = 11, P (2) is true and 7(2) + 1 = 15 thus Q (2) is false. Therefore P (2) = Q (2) is false. (c). If 5(6) + 1 is prime, then 7(6) + 1 is prime. Since 5(6) + 1 = 31, P (5) is true and 7(6) + 1 = 43 thus Q (7) is true. Therefore P (7) = Q (7) is true. (# 2.20) In each of the following, two open sentences P ( x ) and Q ( x ) over a domain S are given. Determine all x S for which P ( x ) = Q ( x ) is a true statement. (a). P ( x ) : x - 3 = 4; Q ( x ) : x 8; S = R . (b). P ( x ) : x 2 1; Q ( x ) : x 1; S = R . (c). P ( x ) : x 2 1; Q ( x ) : x 1; S = N . (d). P ( x ) : x [ - 1 , 2]; Q ( x ) : x 2 2; S = [ - 1 , 1]. Solution . (a). P ( x ) is true if x = 7 and P ( x ) is false if x = 7. Q ( x ) is true if x 8 and Q ( x ) is false if x < 8. We know that P ( x ) = Q ( x ) is false only if P ( x ) is true and Q ( x ) is false. This only occurs if x = 7. Therefore P ( x ) = Q ( x ) is true for all other x . That is, it is true if x = 7.
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