question for comp sci students

[stickybytes]

i need help with this following question:

Is the set R^2, the ordered pairs of real numbers countable? If so, provide the necessary injective function or a method by which one may count or list all elements of the set. If not, explain why.

Thanks.

 

[jiggahertz]

No it is not countable, for the same reason real numbers are not countable. There is not a one to one correspondance with natural numbers. This can be shown through Cantor's diagonalization theorem.

 

[tfinch2]

1
1.000...01
1.000...02

Nope, not countable.

 

[Matthias99]

For something to be 'countable', you need to be able to form a 1-to-1 correspondence with the 'counting numbers' (the set {1, 2, 3, ... }). It is impossible to enumerate all the real numbers (and therefore also impossible to enumerate all the pairs of real numbers).

As for proving it... look at trying to count all the real numbers in a subset of R.