let f be a function so that f: N → N. Define rec∞(f ) = {n ∈ N | f -1(n) is infinite}, as in, the set of all the naturals such that they have infinite pre-images in f. Show the following: 1. The set X1 = {f : N → N | |rec∞(f)| = 1} is countable. 2. The set X2 = {f : N → N | |rec∞(f)| = 2} isn't countable.

let f be a function so that f: N → N. Define rec∞(f ) = {n ∈ N | f -1(n) is infinite}, as in, the set of all the naturals such that they have infinite pre-images in f. Show the following: 1. The set X1 = {f : N → N | |rec∞(f)| = 1} is countable. 2. The set X2 = {f : N → N | |rec∞(f)| = 2} isn't countable.

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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