Question

Suppose that A and B are denumerable, but not disjoint sets. Prove A U B is countable.

Answer 1

emark? A subset of a denumerable set is either con finite or denumerable. - Also, union of a finite set and a denumerable Set is denumerable. 2 for our problema A and B both are denumerable and not disjoint. We have to prove AUR that is countable, a since A and B are not disjoint then, an to Let A = 5 , S₂ =B A ! then sus₂ = AUB and Singa=0 - Now, sy is a subset of B, so either so is finire or so is denumerable. (remork ) If so is finire , then si usa is the union of a denumerable ser and a finite set. Therefore Sus2 is de numerable. Cremark ) If s₂ is denumerable, Let si = 2 a, as,... }, S2 =9bj, by, ...}

Let us de fine a mapping in Suso by f(n) = antot if n is odd - if n is even more then it is a bijeetion and therefore S, US, is denumerable. thus in both case sius, is denumerable Consequently AUB = sus, is countable