Given the following function f:N --> N, define the following set: rec∞(f) = {n ∈ N | f -1(n) is infinite}, as in, the set of all the naturals that have infinite pre-images in f. How to show the following? the set X1 = {f : N → N | |rec∞(f)| = 1 and img(f) is finite} is countable.

Given the following function f:N --> N, define the following set: rec∞(f) = {n ∈ N | f -1(n) is infinite}, as in, the set of all the naturals that have infinite pre-images in f. How to show the following? the set X1 = {f : N → N | |rec∞(f)| = 1 and img(f) is finite} is 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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Given the following function f:N --> N, define the following set: rec∞(f) = {n ∈ N | f ^-1(n) is infinite}, as in, the set of all the naturals that have infinite pre-images in f. How to show the following?

the set X1 = {f : N → N | |rec∞(f)| = 1 and img(f) is finite} is countable.

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