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

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Topic 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 = and xy = for some integers p, q, m, m 9 n and n (with q and n not equal to 0). It follows that y xy E|F|AIS Therefore, xy must be rational. mq np That implies y is rational which is a contradiction. Please answer the following three questions in your post for this module: 1. Find a specific counterexample to show that the theorem is false. 2. Explain what is wrong with the proof. 3. What additional condition on x in the hypothesis would make the conclusion true? Topic 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 = and xy = for some integers p, q, m, m 9 n and n (with q and n not equal to 0). It follows that y xy E|F|AIS Therefore, xy must be rational. mq np That implies y is rational which is a contradiction. Please answer the following three questions in your post for this module: 1. Find a specific counterexample to show that the theorem is false. 2. Explain what is wrong with the proof. 3. What additional condition on x in the hypothesis would make the conclusion true?

Answer rating: 100% (QA)

SOLUTION Sure Id be happy to help 1 Counterexample Let x 2 and y 2 Then x is rational and y is irrational but xy 22 is rational 2 Whats wrong with the ... View the full answer