MATHEMATICS 271 L02 WINTER 2014
Thursday, January 30, 2014 Duration: 30 minutes
 1. Write the negation of each of the following statements. Note that the answer \It is
not the case that:::" is not acceptable.
(a) There exists an integer n such that n + n is odd.
Solution: For every integer n, n + n is even.
(b) For all real numbers x, there exists a real number y such that bx+yc = bxc+byc.
Solution: There exists a real number x such that for every real number y,
bx + yc =6 bxc + byc.
(c) For all integers a;b, and c, if a is prime and ajbc, then ajb or ajc.
Solution: There exist integers a;b, and c so that a is prime and ajbc and
a6 jb and a6 jc.
(d) There exists an integer m such that for all integers n, m + n is even.
Solution: For every integer m there exists and integer n such that m + n
(e) For every positive real number ", there exists a rational number r such that for
every integer m, jm ▯ rj < ".
Solution: There exists a positive real number " such that for every rational
number r there exists an integer m such that jm ▯ rj ▯ ". 2