Kenneth Rosen Edition 7 Exercise 2.5 Question 38 (Page No. 177)

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} \dots d_{n}\dots $ the function $f$ with $f (n) = dn.]$

Answer 1

SEE HERE

LETS NAME THE SETS

A=Z+(POSITIVE INT)  B={0,1,2,3,4,5,6,7,8,9}

NO OF FXN POSSIBLE BETWEEN A AND B IS

1O^Z+(10 KI POWER Z+)

NOW SEE A THEOREM  2^(COUNTABLE INFINITE)=UNCOUNTABLE INFINITE

HERE Z+ IS COUNTABLE INFINITE BUT 2^Z+ IS UNCOUNTABLE INFINITE