Let S be an infinite set, and suppose that f: S Nis an injection. Show that S is a countable set. Let S = (2,3, 5, 7,.} be the set of all prime numbers, and let P(S) = {T |TC S} be the power set of S. (1) Show that P(S) is uncountable. Next, let F(S) = {T |TC S and T is finite) be the set of all finite subsets of S, and define the function /: F(S) → N by setting /({p.P2..Pk}) = PP2 . Pk for each non-empty set of k primes, (k e N), as well as f(ø) = 1. (i) Show that F(S) is an infinite set, and that f is injective but not surjective. (i) Is F(S) uncountable? Justify your answer.
Let S be an infinite set, and suppose that f: S Nis an injection. Show that S is a countable set. Let S = (2,3, 5, 7,.} be the set of all prime numbers, and let P(S) = {T |TC S} be the power set of S. (1) Show that P(S) is uncountable. Next, let F(S) = {T |TC S and T is finite) be the set of all finite subsets of S, and define the function /: F(S) → N by setting /({p.P2..Pk}) = PP2 . Pk for each non-empty set of k primes, (k e N), as well as f(ø) = 1. (i) Show that F(S) is an infinite set, and that f is injective but not surjective. (i) Is F(S) uncountable? Justify your answer.
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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Transcribed Image Text:(c) Let S be an infinite set, and suppose that f : S Nis an injection. Show that S is a countable
set.
(d) Let S = {2, 3, 5, 7, ...} be the set of all prime numbers, and let P(S) = {T |TC S} be the
power set of S.
() Show that P(S) is uncountable.
Next, let F(S) = {T |TC S and T is finite} be the set of all finite subsets of S, and define the
function f : F(S) → N by setting f({p. PP}) Pip2... Pk for each non-empty set of
k primes, (k e N), as well as f(Ø) = 1.
(i) Show that F(S) is an infinite set, and that f is injective but not surjective.
(iii) Is F(S) uncountable? Justify your answer.
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