Countability subset of the reals proof

[hlin818]

Homework Statement

Let (a,b)=XUY, X,Y arbitrary sets where (a,b) is an arbitrary interval. Prove that either X or Y has the same cardinality as that of (a,b).

Homework Equations

The Attempt at a Solution

Really lost.

 

[lanedance]

first show (a,b) is uncountable this shows XUY must be uncountable, now assume X & Y are both countable and look for a contradiction

 

[hlin818]

first show (a,b) is uncountable this shows XUY must be uncountable, now assume X & Y are both countable and look for a contradiction

If we do this by contradiction wouldn't the negation be to assume that X and Y do not have cardinality equal to (a,b)?

 

[lanedance]

no, you wnat a ssume that neither has cardinality equivalent to an interval and find a contradiction that says it can't be so