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(a) Prove that Vr € R, ze Q⇒ (x + 1) E Q. Hint: go back to the definition of Q, and show that if a satisfies that definition, then so does (x + 1). (b) Prove that Vr € R, z Q⇒ (x + 1) # Q. Hint: can you prove the contrapositive of this (c) statement? Prove that Vr € R, z² + 6x +8.5 = 0⇒zQ. Hint: you may want to use results that are similar to 2a and 2b. C

(a) Prove that Vr € R, ze Q⇒ (x + 1) E Q. Hint: go back to the definition of Q, and show that if a satisfies that definition, then so does (x + 1). (b) Prove that Vr € R, z Q⇒ (x + 1) # Q. Hint: can you prove the contrapositive of this (c) statement? Prove that Vr € R, z² + 6x +8.5 = 0⇒zQ. Hint: you may want to use results that are similar to 2a and 2b. C

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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2. For this question, you will prove that the roots of z² + 6x + 8.5 are irrational. In other words, that Va € R, z² + 6x + 8.5=0=zQ. You may use, without proof, the quadratic formula. In other words, you may use the fact that for all real a, b, and c, -b-√b²-4ac, -b+√b²-4ac Vr € R, (ar²+bx+c=0) ^ (b²-4ac > 0) ⇒ (z = -) V (x= 2a 2a Please write complete proofs in each subquestion, without referring back to earlier subquestions. 1 (a) Prove that VI ER, re Q⇒ (r+1) EQ. Hint: go back to the definition of Q, and show that if a satisfies that definition, then so does (x + 1). (b) Prove that Vr € R, z Q⇒ (x + 1) #Q. Hint: can you prove the contrapositive of this statement? (c) Prove that Vr € R, z² + 6x +8.5 = 0⇒xQ. Hint: you may want to use results that are similar to 2a and 2b.
Transcribed Image Text:2. For this question, you will prove that the roots of z² + 6x + 8.5 are irrational. In other words, that Va € R, z² + 6x + 8.5=0=zQ. You may use, without proof, the quadratic formula. In other words, you may use the fact that for all real a, b, and c, -b-√b²-4ac, -b+√b²-4ac Vr € R, (ar²+bx+c=0) ^ (b²-4ac > 0) ⇒ (z = -) V (x= 2a 2a Please write complete proofs in each subquestion, without referring back to earlier subquestions. 1 (a) Prove that VI ER, re Q⇒ (r+1) EQ. Hint: go back to the definition of Q, and show that if a satisfies that definition, then so does (x + 1). (b) Prove that Vr € R, z Q⇒ (x + 1) #Q. Hint: can you prove the contrapositive of this statement? (c) Prove that Vr € R, z² + 6x +8.5 = 0⇒xQ. Hint: you may want to use results that are similar to 2a and 2b.
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