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Let S be the statement: The cube root of every irrational number is irrational. This statement is true, but the following "proof" is incorrect. Ex- plain the mistake. "Proof (by contradiction): Suppose not. Suppose the cube root of every irrational number is rational. But 2√2 is irrational because it is a product of a rational and an irrational number, and the cube root of 2√2 is √2, which is irratio- nal. This is a contradiction, and hence it is not true that the cube root of every irrational number is rational. Thus the statement to be proved is true."

Let S be the statement: The cube root of every irrational number is irrational. This statement is true, but the following "proof" is incorrect. Ex- plain the mistake. "Proof (by contradiction): Suppose not. Suppose the cube root of every irrational number is rational. But 2√2 is irrational because it is a product of a rational and an irrational number, and the cube root of 2√2 is √2, which is irratio- nal. This is a contradiction, and hence it is not true that the cube root of every irrational number is rational. Thus the statement to be proved is true."

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. Let S be the statement: The cube root of every irrational number is irrational. This statement is true, but the following "proof” is incorrect. Ex- plain the mistake. "Proof (by contradiction): Suppose not. Suppose the cube root of every irrational number is rational. But 2√2 is irrational because it is a product of a rational and an irrational number, and the cube root of 2√2 is √2, which is irratio- nal. This is a contradiction, and hence it is not true that the cube root of every irrational number is rational. Thus the statement to be proved is true."
Transcribed Image Text:5. Let S be the statement: The cube root of every irrational number is irrational. This statement is true, but the following "proof” is incorrect. Ex- plain the mistake. "Proof (by contradiction): Suppose not. Suppose the cube root of every irrational number is rational. But 2√2 is irrational because it is a product of a rational and an irrational number, and the cube root of 2√2 is √2, which is irratio- nal. This is a contradiction, and hence it is not true that the cube root of every irrational number is rational. Thus the statement to be proved is true."
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