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a) Prove that √5 is irrational by giving a proof by contradiction.

a) Prove that √5 is irrational by giving a proof by contradiction.

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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Question 2: Using proof methods, prove statements below. (Solve at least 8 proofs (2 from each method)) a) Prove that √5 is irrational by giving a proof by contradiction. b) Prove that if n is an integer and 3n+ 2 is even, then n is even using a proof by contradiction. c) Prove that for any integer x, the integer x(x + 1) is even. (direct) d) Prove n < 2", VnZ+ (induction) 1³ +2³+ + n³ = n²(n+1)² e) (induction) 4 f) Prove that if p; q are positive integers such that plq and qlp, then p = q. (direct) g) Use a direct proof to show that if is an odd integer number, then x² = 8m + 1 Where m is integer. h) Give a proof by cases that for integers x, y, we have |xy| = |x|ly|. i) Prove that |x-1| + x + 5| ≥ 6 for all real numbers x. (by cases) j) The expression 2m² - 1 is odd for all integers m. (by cases) k) Prove that i=1 1≤2√n. (induction) 1) There exist no integers a and b for which 21a + 30b = 1. (contradiction)
Transcribed Image Text:Question 2: Using proof methods, prove statements below. (Solve at least 8 proofs (2 from each method)) a) Prove that √5 is irrational by giving a proof by contradiction. b) Prove that if n is an integer and 3n+ 2 is even, then n is even using a proof by contradiction. c) Prove that for any integer x, the integer x(x + 1) is even. (direct) d) Prove n < 2", VnZ+ (induction) 1³ +2³+ + n³ = n²(n+1)² e) (induction) 4 f) Prove that if p; q are positive integers such that plq and qlp, then p = q. (direct) g) Use a direct proof to show that if is an odd integer number, then x² = 8m + 1 Where m is integer. h) Give a proof by cases that for integers x, y, we have |xy| = |x|ly|. i) Prove that |x-1| + x + 5| ≥ 6 for all real numbers x. (by cases) j) The expression 2m² - 1 is odd for all integers m. (by cases) k) Prove that i=1 1≤2√n. (induction) 1) There exist no integers a and b for which 21a + 30b = 1. (contradiction)
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