The discussion revolves around proving a property of the union of two arbitrary sets, X and Y, within the context of the interval (a,b) on the real number line. The goal is to demonstrate that at least one of the sets X or Y has the same cardinality as the interval (a,b).
There are varying interpretations of how to approach the proof, particularly regarding the assumptions made about the cardinalities of X and Y. Some participants suggest a contradiction method, while others clarify the nature of the assumptions that should be made to reach a contradiction.
Participants are navigating the implications of cardinality and the definitions of countable versus uncountable sets, with some uncertainty about the correct assumptions to make in the proof process.
lanedance said:first show (a,b) is uncountable this shows XUY must be uncountable, now assume X & Y are both countable and look for a contradiction