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Unformatted text preview: SOLUTIONS TO HOMEWORK 5 MATH 150, FALL 09 Problem 1. Section 2.2/ Exercise 8. Suppose is a set of formulas such that for all sentences , either | = or | = . Assume that A | = . Show that for any , A | = iff | = . For one of the directions, suppose that | = . Then by definition of logical implication and since A | = , it follows that A | = . For the other direction, suppose that A | = . Then A 6| = . By the definiton of logical implication and since A | = , it follows that 6| = . So, by the assumptions on , we get that | = . Problem 2. Section 2.2/ Exercise 9. Assume that the language has equality and a two-place predicate P . For each of the following find a sentence such that the model A satisfies iff the condition is met. (1) The universe of A has exactly two members. (2) P A is a function (from the universe of A to itself). (3) P A is a permutation of the universe of A (i.e. it is a one-to-one, onto function). (1) = x y ( ( x = y ) z ( z = x z = y )) (2) = x y ( P ( x,y ) z ( P ( x,z ) y = z )) (3) Let f = x y ( P ( x,y ) z ( P ( x,z ) y = z )) and 11 = x y ( z...
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