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CMSC 250 Quiz #6 ANSWERS Wednesday, Oct. 9, 2002
Write all answers legibly in the space provided. The number of points possible for each question is indicated
in square brackets { the total number of points on the quiz is 30, and you will have exactly 20 minutes to
complete this quiz. You may not use calculators, textbooks or any other aids during this quiz.
1. [30 pnts.] Disprove by counter example or Prove each of the following:
a. The quotient of any two positive rationals is rational.
ANSWER - true - proof by generalizing from the generic particular:
8x; y 2 Q+; xy 2 Q
Proof:
Let p and q be arbitrary in Q.
9a; b; c; d 2 Z; p = ab ^ q = cd ^ b 6= 0 ^ d 6= 0 by de nition of rational.
a > 0 ^ c > 0 ^ b > 0andd > 0 by de nition of positive in algebra.
p
q =
a
bc
d
by substitution.
= ab dc = adbc by algebra.
ad; bc 2 Z by closure of Z during multiplication.
bc 6= 0 by algebra.
Therefore, pq 2 Q by de nition of rational.
8x; y 2 Q+; xy 2 Q by generalizing from the generic particular.
b. ...

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