MATH 266 (091) ASSIGNMENT 2
due on Friday, Sept. 12
1. Translate the following English sentences into symbolic sentences with quantifiers
(universe is in parentheses.)
(a) Everybody loves someone. (all people)
(b) Some people are happy and some people are not happy. (all people)
(c) All differentiable functions are continuous. (all functions)
(d) Every finite subset of natural numbers has a largest element. (all subsets of
natural numbers)
(e) For every positive real number x, there is a unique real number ysuch that
ey=x. (all real numbers)
2. Let xand ybe integers. Prove that if xand yare even, then xy is divisible by
4.
3. Let aand bbe real numbers. Use the method of “proof by exhaustion” (see page
36) to prove that |a−b|=|b−a|.
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