Solved: Fill in the blanks for the following proof that

Chapter 4, Problem 4.231

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Fill in the blanks for the following proof that the difference of any rational number and any irrational number is irrational. Proof: Suppose not. That is, suppose that there exist (a) x and (b) y such that x y is rational. By denition of rational, there exist integers a, b, c, andd with b =0 and d =0sothatx = (c) and x y = (d) .Bysubstitution, a b y = c dAdding y and subtractingc don both sides givesy =(e) = ad bd bc bd=adbc bdby algebra.Now both adbc and bd are integers because products and differences of (f) are (g) . Andbd =0 by the (h) . Hence y is a ratio of integers with a nonzero denominator, and thus y is (i) by denition of rational. We therefore have both that y is irrational and that y is rational, which is a contradiction. [Thus the supposition is false and the statement to be proved is true.]