Let T be the statement: The sum of any two rational numbers is rational. T is true, but the following proposed proof by contradiction is incorrect. Proposed proof: Suppose not. That is, suppose that the sum of any two rational numbers is not rational. This means that no matter what two rational numbers are chosen, their sum is not rational. Now, both 1 and 3 are rational because 1 = 1 and 1 3 = and so both are ratios of integers with a nonzero denominator. Hence, by supposition, the sum of 1 and 3 is not rational. But 1 + 3 = 4, and 4 is rational because 4 = 4 which is a ratio of integers with a nonzero denominator. Hence, 4 is both rational and not rational, which is a contradiction. This contradiction shows that the suppoſlition is false, and hence statement T is true. Identify the error(s) in the proposed proof. 3 4 O The error is that 1 which does not have a terminating decimal representation and is therefore not rational. + %3D 1 O The error is that the negation for T that was used in the proposed proof is incorrect. O The error is that 4 is both a rational number and an irrational number so there is no contradiction. O The error is that 4 not %3D 1

Let T be the statement: The sum of any two rational numbers is rational. T is true, but the following proposed proof by contradiction is incorrect. Proposed proof: Suppose not. That is, suppose that the sum of any two rational numbers is not rational. This means that no matter what two rational numbers are chosen, their sum is not rational. Now, both 1 and 3 are rational because 1 = 1 and 1 3 = and so both are ratios of integers with a nonzero denominator. Hence, by supposition, the sum of 1 and 3 is not rational. But 1 + 3 = 4, and 4 is rational because 4 = 4 which is a ratio of integers with a nonzero denominator. Hence, 4 is both rational and not rational, which is a contradiction. This contradiction shows that the suppoſlition is false, and hence statement T is true. Identify the error(s) in the proposed proof. 3 4 O The error is that 1 which does not have a terminating decimal representation and is therefore not rational. + %3D 1 O The error is that the negation for T that was used in the proposed proof is incorrect. O The error is that 4 is both a rational number and an irrational number so there is no contradiction. O The error is that 4 not %3D 1

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

See similar textbooks

Similar questions

Prove that if is a nonzero rational number and is irrational, then is irrational.
If x and y are positive rational numbers, prove that there exists a positive integer n such that nxy. This property is called the Archimedean Property of the rational numbers. (Hint: Write x=a/b and y=c/d with each of a,b,c,d+.)
Prove that if a is rational and b is irrational, then a+b is irrational.
13. Prove that if and are rational numbers such that then there exists a rational number such that . (This means that between any two distinct rational numbers there is another rational number.)
Prove or disprove each of the following statements. The set Q of rational numbers is an ideal of the set R of real numbers. The set Z of integers is an ideal of the set Q of rational numbers.
Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation can be proved to be true for n=k+1. Explain why this does not prove that the statement is true for all positive integers. Is the statement true for all positive integers? Why?
Prove that is irrational. (That is, prove there is no rational number such that .)
Prove by induction that if r is a real number where r1, then 1+r+r2++rn=1-rn+11-r
Use the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such that a2=3b2. Explain how this proves that 3 is not a rational number.