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MATH 266 (092) ASSIGNMENT 2 Due Monday, Feb. 2. 1. Translate the following English sentences into symbolic sentences with quantifiers (universe is in parentheses.) (a) Everybody loves someone. (people) (b) All differentiable functions are continuous. (functions) (c) Every subset of natural numbers has a smallest element. natural numbers) (subsets of (d) For every positive real number x, there is a unique real number y such that ey = x. (real numbers) (e) Between any two real numbers, there is a rational number. (real numbers) 2. Let x and y be integers. Prove that if x and y are even, then xy is divisible by 4. 3. Let a and b be real numbers. Use the method of cases (the author calls it “proof by exhaustion”) to prove that |ab| = |a| |b|. 4. Do Exercise 9(a), on page 38. 1