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Section answers, 4 April 2011 Julian Jonker 1 How much do you know? 1. (Ex 9.25) Assume you are working in an extension of the ﬁrst-order language of arithmetic with the additional predicates Even(x) and Prime(x) . Express the following in this language, explicitly using the function symbol × , as in z × z , rather than z 2 . Note that you do not have predicate Square(x) or Odd(x) . (a) No square is prime. ¬∃ x Prime(x × x) (b) Some square is odd. x ¬ Even(x × x) (c) The square of any prime is prime. x(Prime(x) Prime(x × x)) (d) The square of any prime other than 2 is odd. x(Prime(x) x > 2 ) → ¬ Even(x × x)) (e) The square of any number greater than 1 is greater than the number itself. x(x > 1 x × x > x) 2. Assess whether the following arguments are (a) tautologically valid, (b) logically valid but not tautologically valid, or (c) invalid. (a)

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