Question

Give an example of two uncountable sets A and B with a nonempty intersection, such that A−B is

(a) Finite

(b) Countably infinite

(c) Uncountably infinite

Answer 1

(a)   Let A and B be the same set (A can be any set), then A – B will be a null set which is a finite set.

(b)  Let A be the set of R and B be the set R – Z^–. Then A – B will be the set containing all negative integers which is countable finite.

(c)   A = {x | x $>$ 0, x $\in$ R} and B = {x | x $>$ 5, x $\in$ R}.

Then, A – B = {x | 0 $<$ x $\le$ 5, x $\in$ R} which is countable infinite set.