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Problem 12.2.8. Suppose X is an uncountable set and Y C X is countably infinite. Prove that X and X – Y have the same cardinality.

Problem 12.2.8. Suppose X is an uncountable set and Y C X is countably infinite. Prove that X and X – Y have the same cardinality.

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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Real math analysis, Please help me solve problem 12.2.8

Problem 12.2.8. Suppose X is an uncountable set and Y C X is countably infinite. Prove that X and X – Y have the same cardinality. v Hint. Let Y = Yo. If X – Y, is an infinite set, then by the previous problem it contains a countably infinite set Y1. Likewise if X – (Y, U Y1) is infinite it also contains an infinite set Y2. Again, if X – (Yo UYUY2) is an infinite set then it contains an infinite set Y3, etc. For n = 1,2, 3, ..., let fn : Yn-1 → Yn be a one- to-one correspondence and define f : X → X – Y by f(x) = x, S fn(x), if æ E Yn, n = 0, 1, 2, ... if æ e X – (U,Yn) Show that f is one-to-one and onto. The above problems say that R, T – U, T, and P(N) all have the same cardinality
Transcribed Image Text:Problem 12.2.8. Suppose X is an uncountable set and Y C X is countably infinite. Prove that X and X – Y have the same cardinality. v Hint. Let Y = Yo. If X – Y, is an infinite set, then by the previous problem it contains a countably infinite set Y1. Likewise if X – (Y, U Y1) is infinite it also contains an infinite set Y2. Again, if X – (Yo UYUY2) is an infinite set then it contains an infinite set Y3, etc. For n = 1,2, 3, ..., let fn : Yn-1 → Yn be a one- to-one correspondence and define f : X → X – Y by f(x) = x, S fn(x), if æ E Yn, n = 0, 1, 2, ... if æ e X – (U,Yn) Show that f is one-to-one and onto. The above problems say that R, T – U, T, and P(N) all have the same cardinality
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