Consider the following statement. 3/2 7 is irrational. This statement is true, but the following proposed proof by contradiction is incorrect. Proposed proof: Suppose not. That is, suppose 3 2 – 7 is rational. By definition of rational, there exist real numbers a and b with 3/2 - 7 = 4 and b = 0. Using algebra to solve for 2 gives a + 7 BbV2 - 7b = a; and so v2 3b But a + 7b and 3b are real numbers and 3b + 0. Therefore, by definition of rational, v 2 is rational. This contradicts Theorem 4.7.1, which states that 2 is irrational. Hence the supposition is false. Identify the errors in the proposed proof. (Select all that apply.) O There is an error in the algebra solving for v2. O For a proof by contradiction you should suppose that 32 - 7 is irrational. O The square root of 2 is rational, so there is no contradiction. O To apply the definition of rational, a and b must be integers. O The statement to be proved is assumed.

Consider the following statement. 3/2 7 is irrational. This statement is true, but the following proposed proof by contradiction is incorrect. Proposed proof: Suppose not. That is, suppose 3 2 – 7 is rational. By definition of rational, there exist real numbers a and b with 3/2 - 7 = 4 and b = 0. Using algebra to solve for 2 gives a + 7 BbV2 - 7b = a; and so v2 3b But a + 7b and 3b are real numbers and 3b + 0. Therefore, by definition of rational, v 2 is rational. This contradicts Theorem 4.7.1, which states that 2 is irrational. Hence the supposition is false. Identify the errors in the proposed proof. (Select all that apply.) O There is an error in the algebra solving for v2. O For a proof by contradiction you should suppose that 32 - 7 is irrational. O The square root of 2 is rational, so there is no contradiction. O To apply the definition of rational, a and b must be integers. O The statement to be proved is assumed.

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter11: Rational And Irrational Numbers

Section11.1: Properties Of Rational Numbers

Problem 36WE

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