Answer :
Hence Proved N to N is uncountable set by using a diagonalization argument
What is Diagonalization Argument?
Georg Cantor published the Cantor's diagonal argument in 1891 as a mathematical demonstration that there are infinite sets that cannot be put into one-to-one correspondence with the infinite set of natural numbers. It is also known as the diagonalization argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof.
Proof:
We write f as the sequence of value it generates
that is, say f:N-N is defined as f(x) =x then
we write f as : 1,2,3,4........
similarly if f:N-N were f(x) = 2x then we write it as :2,4,6,8.......
now assume on the contrary that the set of functions from N to N is countable
then,
[tex]f_{1}[/tex] : [tex]f_{11}[/tex] [tex]f_{12}[/tex] [tex]f_{13}[/tex] , ...........
[tex]f_{2}[/tex] : [tex]f_{21}[/tex] [tex]f_{22}[/tex] [tex]f_{23}[/tex] , ...........
[tex]f_{3}[/tex] : [tex]f_{31}[/tex] [tex]f_{32}[/tex] [tex]f_{33}[/tex] , ........... and so on
Here [tex]f_{ij}[/tex] =[tex]f_{i}(j)[/tex]
consider the function f as :
f : ([tex]{f11}[/tex] +1 ) , ([tex]{f22}[/tex] +1) , ([tex]{f33}[/tex] +1),.........
Then f wouldn't appear in the list of the function we had above since any [tex]f_{i}[/tex] would disagree with f at the i th place
Therefore the set of the function from N to N is an uncountable set
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