Fill in the blanks for the following proof that the difference of any rational number and...

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 definition of rational, there exist integers a, b, c, and d with b ≠ 0 and d ≠ 0 so that x = (c) and x −y = (d). By substitution,

Adding y and subtracting  on both sides gives

 by algebra.

Now both ad − bc and bd are integers because products and differences of (f) are (g). And bd ≠ 0 by the (h). Hence y is a ratio of integers with a nonzero denominator, and thus y is (i) by definition 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.]