# hw06 - Q(the rationals(b Domain is I(the irrationals(c...

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Spring-2011 CMSC 250: Homework 6 Due: Mon Mar 7, 2011 NOTE: Due AT THE BEGINNING of Recitation! COURSE WEBSITE: http://www.cs.umd.edu/ gasarch/250/S11.html (NOTE- there is a Tilda before gasarch. It may be easier to get to the course website via the dept webpage (go to class webpages) or gasarch’s homepage (it will be obvious).) 1. (10 points) What is your name? Write it clearly. Staple your HW. WHERE and WHEN are you taking the midterm? (SEE shortsyll to see WHERE.) If you cannot make the midterm time and have NOT been in contact with Dr. Gasarch about it then contact him ASAP 2. (30 points) Consider the sentence ( x )( y ) ± b x c + d y e ≤ d x e + b y c ² . For each of the following domains say if the sentence is TRUE or FALSE when the quantiﬁers range over those domains. Prove your result. (a) Domain is
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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 â‰¡ (mod 5) then x â‰¡ (mod 5). (b) Prove that if 2 x 2 â‰¡ (mod 5) then x â‰¡ (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) ( âˆ€ x ) Â± d x e < x + 1 Â² . (b) ( âˆ€ x ) Â± 5 / 4 < x < 3 / 2 â†’ x 2 6 = b x c Ã— d x e Â² . (c) ( âˆ€ x ) Â± 3 / 2 < x < 7 / 4 â†’ x 2 6 = b x c Ã— d x e Â² . (d) ( âˆ€ x )( âˆƒ y ) Â± y > x âˆ§ b x c 6 = d x e Â² . 1...
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