Show that the set R Q of all irrational numbers is not denu

Show that the set R - Q of all irrational numbers is not denumerable.

Solution

Solution:

we know that Q is denumerable.

Proof by contradiction.

Let R-Q is denumerable.Then (R-Q) union Q is also denumerable since,union of two denumerable sets is also denumerable.

This imply R is denumerable - a contradiction.

Therefore R-Q is not denumerable.