CPSC 233 Study Guide - Quiz Guide: Rational Number

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[5: 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 n2 + n is odd. Solution: for every integer n, n2 + n is even. (b) for all real numbers x, there exists a real number y such that (cid:98)x+y(cid:99) = (cid:98)x(cid:99)+(cid:98)y(cid:99). Solution: there exists a positive real number such that for every rational number r there exists an integer m such that |m r| . [5: prove that for all real numbers r and s, if r and r + s are rational, then s is rational. Let r and s be real numbers, and assume that r and r+s are rational numbers. Then there exist integers a, b, c, and d with b, d (cid:54)= 0 such that r = a/b and r+s = c/d. By substitution, we have c d a b.