Solution Found!
Show that the set of functions from the positive integers
Chapter 2, Problem 38E(choose chapter or problem)
Show that the set of functions from the positive integers to the set \(\{0,1,2,3,4,5,6,7,8,9\}\) is uncountable. [Hint: First set up a one-to-one correspondence between the set of real numbers between 0 and 1 and a subset of these functions. Do this by associating to the real number \(0 . d_{1} d_{2} \ldots d_{n} \ldots\) the function \(f\) with \(\left.f(n)=d_{n} .\right]\)
Equation Transcription:
Text Transcription:
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
0.d_1d_2 . . . dn . . .
f
f (n) = d_n.]
Questions & Answers
QUESTION:
Show that the set of functions from the positive integers to the set \(\{0,1,2,3,4,5,6,7,8,9\}\) is uncountable. [Hint: First set up a one-to-one correspondence between the set of real numbers between 0 and 1 and a subset of these functions. Do this by associating to the real number \(0 . d_{1} d_{2} \ldots d_{n} \ldots\) the function \(f\) with \(\left.f(n)=d_{n} .\right]\)
Equation Transcription:
Text Transcription:
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
0.d_1d_2 . . . dn . . .
f
f (n) = d_n.]
ANSWER:
SOLUTION
Step 1
We have to show that the set of functions from the set of positive integers to the set
{ 0,1,2,3,4,5,6,7,8,9} is uncountable.