Proof that the irrational numbers are uncountable

Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set Q of rational numbers is countable, but how would you show that the irrationals are uncountable?

Answer

Given that the reals are uncountable (which can be shown via Cantor diagonalization) and the rationals are countable, the irrationals are the reals with the rationals removed, which is uncountable. (Or, since the reals are the union of the rationals and the irrationals, if the irrationals were countable, the reals would be the union of two countable sets and would have to be countable, so the irrationals must be uncountable.)

Attribution
Source : Link , Question Author : nkassis , Answer Author : J. M. ain’t a mathematician

Leave a Comment