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(b) Evaluate whether the following argument is correct; if not, then specify which lines are incor- rect steps in the reasoning. Each line is assessed as if the other lines are all correct. So, you are to identify which lines (the minimum number) would you need to fix to get a correct proof. Proposition: If r and y are rational numbers then 3x + 2y is also a rational number. Proof: 1. We proceed by contradiction proof. 2. Assumer and y are irrational numbers. 3. Since z and y are rational, z = and y = 5, where a, b, c, and d are integers, and b 4. We will show that 3x + 2y is a rational number. 0 and d = 0. 3ad+2bc bd = 5. Plugging in for r and for y into the expression 3x +2y gives: 3x + 2y = 3 + 2 6. Since a, b, c, and d are all integers, 3ad +2be and bd are also integers. 7. Since b = 0 and d + 0, bd + 0. 8. Therefore, 3ad + 2be and bd contradict the assumption that z and y are irrational numbers, which implies that 3x + 2y is irrational is false.

(b) Evaluate whether the following argument is correct; if not, then specify which lines are incor- rect steps in the reasoning. Each line is assessed as if the other lines are all correct. So, you are to identify which lines (the minimum number) would you need to fix to get a correct proof. Proposition: If r and y are rational numbers then 3x + 2y is also a rational number. Proof: 1. We proceed by contradiction proof. 2. Assumer and y are irrational numbers. 3. Since z and y are rational, z = and y = 5, where a, b, c, and d are integers, and b 4. We will show that 3x + 2y is a rational number. 0 and d = 0. 3ad+2bc bd = 5. Plugging in for r and for y into the expression 3x +2y gives: 3x + 2y = 3 + 2 6. Since a, b, c, and d are all integers, 3ad +2be and bd are also integers. 7. Since b = 0 and d + 0, bd + 0. 8. Therefore, 3ad + 2be and bd contradict the assumption that z and y are irrational numbers, which implies that 3x + 2y is irrational is false.

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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Q4 ( b) part
(b) Evaluate whether the following argument is correct; if not, then specify which lines are incor- rect steps in the reasoning. Each line is assessed as if the other lines are all correct. So, you are to identify which lines (the minimum number) would you need to fix to get a correct proof. Proposition: If it and y are rational numbers then 3x + 2y is also a rational number. Proof: 1. We proceed by contradiction proof. 2. Assume and y are irrational numbers. 3. Since r and y are rational, z = and y = 5, where a, b, c, and d are integers, and b 4. We will show that 3x + 2y is a rational number. 0 and d 0. = 5. Plugging in for r and for y into the expression 3x +2y gives: 3x + 2y = 3 + 2 6. Since a, b, c, and d are all integers, 3ad +2be and bd are also integers. 7. Since b 0 and d 0, bd + 0. 3ad+2bc bd 8. Therefore, 3ad + 2be and bd contradict the assumption that r and y are irrational numbers, which implies that 3x + 2y is irrational is false.
Transcribed Image Text:(b) Evaluate whether the following argument is correct; if not, then specify which lines are incor- rect steps in the reasoning. Each line is assessed as if the other lines are all correct. So, you are to identify which lines (the minimum number) would you need to fix to get a correct proof. Proposition: If it and y are rational numbers then 3x + 2y is also a rational number. Proof: 1. We proceed by contradiction proof. 2. Assume and y are irrational numbers. 3. Since r and y are rational, z = and y = 5, where a, b, c, and d are integers, and b 4. We will show that 3x + 2y is a rational number. 0 and d 0. = 5. Plugging in for r and for y into the expression 3x +2y gives: 3x + 2y = 3 + 2 6. Since a, b, c, and d are all integers, 3ad +2be and bd are also integers. 7. Since b 0 and d 0, bd + 0. 3ad+2bc bd 8. Therefore, 3ad + 2be and bd contradict the assumption that r and y are irrational numbers, which implies that 3x + 2y is irrational is false.
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