# Prove that {0,1}^N is uncountable.

## Homework Statement

Prove that {0,1}N is uncountable.

(N = positive integers)

f: N -> {0,1}

## The Attempt at a Solution

f(n) = 0 or 1 for every n is an element of N.
I should treat n as the nth term of an infinitely long sequence of 0's and 1's.

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LCKurtz
Homework Helper
Gold Member
Are you familiar with Cantor's diagonal argument which shows the reals are uncountable?

Yes, we discussed that today in class, but I'm a bit confused on Cantor's argument. We also discussed possibly proving the statement through contradiction, assuming that it is countable.

HallsofIvy
Homework Helper
Every real number, between 0 and 1, can be written in binary notation as a list of "0"s and "1"s so your set is equivalent to the set of all real numbers between 0 and 1.

LCKurtz
Homework Helper
Gold Member
Yes, we discussed that today in class, but I'm a bit confused on Cantor's argument. We also discussed possibly proving the statement through contradiction, assuming that it is countable.
Sounds like a good plan. Kind'a like Cantor's argument...

Can you explain to me what Cantor's argument is though?

LCKurtz