2. Let x, y be real numbers. Determine which of the following statements are true. For those that are true, prove them. For those that are false, provide a counterexample. For the purpose of a counterexample you may assume without proof that 2, e, 7 and In 2 are all irrational. (a) If x and y are both rational, then xy is rational. (b) If x is rational and y is irrational, then xy is rational. (c) If x is rational and y is irrational, then xy is irrational. (d) If x and y are both irrational, then xy is irrational.

2. Let x, y be real numbers. Determine which of the following statements are true. For those that are true, prove them. For those that are false, provide a counterexample. For the purpose of a counterexample you may assume without proof that 2, e, 7 and In 2 are all irrational. (a) If x and y are both rational, then xy is rational. (b) If x is rational and y is irrational, then xy is rational. (c) If x is rational and y is irrational, then xy is irrational. (d) If x and y are both irrational, then xy is irrational.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.1: The Field Of Real Numbers

Problem 20E: Give counterexamples for the following statements. If and are irrational, then is irrational. If ...

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