I have a question regarding whether the intersection of the complement of two disjoint set is disjoint or not.
I mean given say $A$ and $B$ with $A$ and $B$ disjoint, i.e. $A \subset X$ and $B \subset X$, and $A \cap B = \emptyset$. It seems that $A^{c} \cap B^{c} \ne \emptyset$, at least when I draw a Venn Diagram, it seems the intersection of the complement is not empty given the condition that $A$ and $B$ are disjoint. But somehow I am having some difficulty proving it.
Could someone give me some hint. Because it seems it is not a very difficulty proof. But I kind of get stuck.
Thank you
Best Answer
I think the answer you proposed need some alterations.
For example, Let: X=set of real numbers $ A = (-\infty , 0 ] \\ $ $ B = (0, \infty )$
So the intersection of the complements are empty.
But the result you proposed is true when A and B are proper subsets of X