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Order 4 of the following sentences so that they prove the following statement by contradiction: The average of three real numbers is greater than or equal to at least one of the numbers. Proof by contradiction of the statement (in order): Choose from this list of sentences Hence, the assumption that the average of three real numbers is greater than all of the numbers is false Adding up the three inequalities yields x+y+z>x+y+z Adding up the three inequalities yields x+y+zx, (x+y+z)/3> y, and (x+y+z)/3> z This contradicts the fact that

Order 4 of the following sentences so that they prove the following statement by contradiction: The average of three real numbers is greater than or equal to at least one of the numbers. Proof by contradiction of the statement (in order): Choose from this list of sentences Hence, the assumption that the average of three real numbers is greater than all of the numbers is false Adding up the three inequalities yields x+y+z>x+y+z Adding up the three inequalities yields x+y+zx, (x+y+z)/3> y, and (x+y+z)/3> z This contradicts the fact that

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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Order 4 of the following sentences so that they prove the following statement by contradiction: The average of three real numbers is greater than or equal to at least one of the numbers. Proof by contradiction of the statement (in order): Choose from this list of sentences Hence, the assumption that the average of three real numbers is greater than all of the numbers is false Adding up the three inequalities yields x+y+z> x+y+z Adding up the three inequalities yields x+y+z<x+y+z Let x, y and z be real numbers and (x+y+z)/3>x, (x+y+z)/3> y, and (x+y+z)/3> < This contradicts the fact that x+y+z= x+y+z Let x, y and z be real numbers and (x+y+z)/3 < x, (x+y+z)/3 < y, and (x+y+z)/3 <z Hence, the assumption that the average of three real numbers is less than all of the numbers is false
Transcribed Image Text:Order 4 of the following sentences so that they prove the following statement by contradiction: The average of three real numbers is greater than or equal to at least one of the numbers. Proof by contradiction of the statement (in order): Choose from this list of sentences Hence, the assumption that the average of three real numbers is greater than all of the numbers is false Adding up the three inequalities yields x+y+z> x+y+z Adding up the three inequalities yields x+y+z<x+y+z Let x, y and z be real numbers and (x+y+z)/3>x, (x+y+z)/3> y, and (x+y+z)/3> < This contradicts the fact that x+y+z= x+y+z Let x, y and z be real numbers and (x+y+z)/3 < x, (x+y+z)/3 < y, and (x+y+z)/3 <z Hence, the assumption that the average of three real numbers is less than all of the numbers is false
Complete the following proof, which proves the statement by contradiction: If a² is even, then a is even. proof. Let a² be By the definition of odd, there exist integers k such that It follows that a² = (2k + 1)² = 4k² + 4k+ 1 = 2( Since P and a be is an integer, and integers are closed under must be an integer. Thus, by the definition of odd, This contradicts with that is this problem is odd. Hence, )+1. and is odd. Hence, the assumption that a² is even and a is odd is false.
Transcribed Image Text:Complete the following proof, which proves the statement by contradiction: If a² is even, then a is even. proof. Let a² be By the definition of odd, there exist integers k such that It follows that a² = (2k + 1)² = 4k² + 4k+ 1 = 2( Since P and a be is an integer, and integers are closed under must be an integer. Thus, by the definition of odd, This contradicts with that is this problem is odd. Hence, )+1. and is odd. Hence, the assumption that a² is even and a is odd is false.
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