Consider the following theorem and proof.Theorem: If x is rational and y is irrational, th...

Consider the following theorem and proof.

Theorem: If x is rational and y is irrational, then xy is irrational.

Proof: Suppose x is rational and y is irrational. If xy is rational, then we have x = p /q and xy = m/n for some integers p, q, m, and n, with q ≠ 0 and n≠ 0. It follows that

This implies that y is rational, a contradiction. We conclude that xy must be irrational.

(a) Find a specific counterexample to show that the theorem is false.

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(b) Explain what is wrong with the proof.

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(c) What additional condition on x in the hypothesis would make the conclusion true?