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Unformatted text preview: Q (the rationals). (b) Domain is I (the irrationals). (c) Domain is Z (the integers). 3. (40 points) (a) Prove that if x 2 (mod 5) then x (mod 5). (b) Prove that if 2 x 2 (mod 5) then x (mod 5). (c) Prove that 10 1 / 2 is not rational WITHOUT using the Unique Factorization Theorem. (Use the two lemmas in parts a and b of this problem.) (d) Prove that 10 1 / 2 is not rational USING the Unique Factorization Theorem. 4. (20 points) For each of the following say if it is TRUE or FALSE and WHY. The domain is the reals. (a) ( x ) d x e < x + 1 . (b) ( x ) 5 / 4 < x < 3 / 2 x 2 6 = b x c d x e . (c) ( x ) 3 / 2 < x < 7 / 4 x 2 6 = b x c d x e . (d) ( x )( y ) y > x b x c 6 = d x e . 1...
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 Spring '08
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