250: Spring-2011
CMSC 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 gasarchs 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
([e-mail])
2. (30 points) Consider the sentence
(x)(y ) x + y x + y .
For each of the following domains say if the sentence is TRUE or FALSE when quantiers
range the over those domains. Prove your result.
(a) Domain is Q (the rationals).
(b) Domain is I (the irrationals).
(c) Domain is Z (the integers).
3. (40 points)
(a) Prove that if x2 0
(b) Prove that if 2x2 0
(mod 5) then x 0
(mod 5) then x 0
(mod 5).
(mod 5).
(c) Prove that 101/2 is not rational WITHOUT using the Unique Factorization Theorem.
(Use the two lemmas in parts a and b of this problem.)
(d) Prove that 101/2 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) x < x + 1 .
(b) (x) 5/4 < x < 3/2 x2 = x x .
(c) (x) 3/2 < x < 7/4 x2 = x x .
(d) (x)(y ) y > x x = x .
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