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109_sp2011_ho_funcns_sol

# 109_sp2011_ho_funcns_sol - 109 Spring 2011 Quantiers and...

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109 Spring 2011 - Quantifiers and Functions - Solutions Exercise (II.11) . Give a proof or a counterexample for each of the following statements. (i) x R y R ( x + y > 0) True: given x R define y = 2 x . (ii) x R y R ( x y > 0) True: given x R define y = 2 + x . (iii) x R y R ( x + y > 0) False. We prove that the negation, x R y R ( x + y 0), is true. Given x , let y = x . (iv) x R y R ( xy > 0) False. Counterexample: x = 0. (v) x R y R ( xy > 0) False. We prove the negation, x R y R ( xy 0). Given x , let y = 0. (vi) x R y R ( xy 0) True: given x R let y = 0. (vii) x R y R ( xy 0) True: choose x = 0. (viii) x R y R ( x + y > 0 or x + y = 0) True: given x R define y = 2 x . (ix) x R y R ( x + y > 0 and x + y = 0) False: the inner predicate can never be made true. (x) ( x R y R ( x + y > 0) ) and ( x R y R ( x + y = 0) ) True: for first conjunct, given x define y = 2 x ; for the second conjunct, given x define y = x .

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109_sp2011_ho_funcns_sol - 109 Spring 2011 Quantiers and...

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