Proposition 2. If H = (x), then |H| = |x| (where if one side of this equality is infinite, so is the other). More specifically (1) if |H| = n < ∞o, then x = 1 and 1, x, x², …, are all the distinct elements of H, and (2) if |H| = ∞o, then x" # 1 for all n ‡ 0 and xª ‡ x¹ for all a + b in Z.
Proposition 2. If H = (x), then |H| = |x| (where if one side of this equality is infinite, so is the other). More specifically (1) if |H| = n < ∞o, then x = 1 and 1, x, x², …, are all the distinct elements of H, and (2) if |H| = ∞o, then x" # 1 for all n ‡ 0 and xª ‡ x¹ for all a + b in Z.
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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Please if able give an example for both propositions showing how they satisfy the given points, especially proposition 2 is the one I struggle with.
Transcribed Image Text:Proposition 2. If H = (x), then |H| = |x| (where if one side of this equality is
infinite, so is the other). More specifically
(1) if |H| = n < ∞o, thenx"= 1 and 1, x, x², …, x-1 are all the distinct elements
of H, and
(2) if |H| = ∞o, then x" # 1 for all n ‡0 and xª #x² for all a ‡ b in Z.
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