Let S be a subset of R". None of these parts should take very much work! (a) Suppose that a > 0. Prove that f: SaS, f (x₁,xn) = (ax₁,..., arn) is a bijection. (b) Suppose that (a₁, a2,...an) ER". Prove that f: SS+ (aan), f(x₁,...,xn) = (x₁+ a₁n+an) is a bijection. (c) You are holding a baseball B in your hand and you are standing on planet E. Prove that |E| = |B]. (Assume that both objects are solid spheres. You do not have to write down any equations-just explain why it is true).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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6. Let S be a subset of R". None of these parts should take very much work!
(a) Suppose that a > 0.
Prove that f: SaS, f (x₁,xn) = (ax₁,...,axn) is a bijection.
(b) Suppose that (a₁, a2,...an) ER".
Prove that f: S→S + (a₁,...,an). f (x₁,xn) = (x₁+ a₁,...,n+an) is a
bijection.
(c) You are holding a baseball B in your hand and you are standing on planet E.
Prove that |E| = |B. (Assume that both objects are solid spheres. You do not
have to write down any equations-just explain why it is true).
Transcribed Image Text:6. Let S be a subset of R". None of these parts should take very much work! (a) Suppose that a > 0. Prove that f: SaS, f (x₁,xn) = (ax₁,...,axn) is a bijection. (b) Suppose that (a₁, a2,...an) ER". Prove that f: S→S + (a₁,...,an). f (x₁,xn) = (x₁+ a₁,...,n+an) is a bijection. (c) You are holding a baseball B in your hand and you are standing on planet E. Prove that |E| = |B. (Assume that both objects are solid spheres. You do not have to write down any equations-just explain why it is true).
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