The Diagonal Method: Proving Sets are Infinite

[Aditya89]

Can anybody tell me the Canter Method of proving that certain sets are infinite? It is called as "Diagonal Method".

 

[Zurtex]

Cantor's diagonal argument was used to show that the natural numbers and the real numbers did not have an equal cardinality, in other words there existed no 1 to 1 correspondence between the two sets.

It talks about it here:

http://mathworld.wolfram.com/CantorDiagonalMethod.html

Or more simply:

http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

 

[Aditya89]

Hey thanks, Zurtex! But from the first link, it does not become clear why it is called "Diagonal Method". Also, can you explain the link between first proof and second proof, please? Also, please tell me how to construct a bijection between Rationals & Reals.

 

[shmoe]

Hey thanks, Zurtex! But from the first link, it does not become clear why it is called "Diagonal Method". Also, can you explain the link between first proof and second proof, please?

See the wiipedia article, step 5 in the proof singles out the 'diagonal' terms of the list. I only count one proof in the links he gave.

Also, please tell me how to construct a bijection between Rationals & Reals.

There isn't one. The diagonal argument shows the reals are uncountable while the rationals are countable.

 

[Aditya89]

Oh! I'm sorry for saying reals & rationals! It's integers and rationals! And why do you count only one proof?

 

[matt grime]

The rationals are famously countable, try constructing the proof yourself. There are two variants. One is anothert kind of diagonal argument, and the other is by remembering that rational numbers are a subset of the pairs of integers.