Prove that irrational numbers have the same cardinality as real numbers

[mathgeek808]

Need help with this proof:
Prove that the set of irrational numbers has the same cardinality as the set of real numbers.

 

[Office_Shredder]

I would do this by a bijection. Pick a countable set A in the irrational numbers (think cosets if you can't come up with an example). Then construct a bijection that is the identity outside of A

 

[Hurkyl]

I would use arithmetic of cardinal numbers myself.

 

[statdad]

The reals are the disjoint union of rationals and irrationals. Take this back to Hurkyl's suggestion.