1
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Consider $A = \{ n \mid n \in \mathbb N \text { and } n \text { is odd } \}$ and $B = \{ m \mid m \in \mathbb N \text { and } m \text { is even } \}$ $A \cap B = \emptyset$ and also $A^C \cap B^C = \emptyset$. $\endgroup$ – Mauro ALLEGRANZA Sep 26 '17 at 12:01
  • $\begingroup$ The "intersection of the complement" is one set, the set $A^c\cap B^c$. What does it mean for one set to be disjoint?? $\endgroup$ – bof Sep 26 '17 at 12:29
0
$\begingroup$

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

$\endgroup$
  • 1
    $\begingroup$ In your counterexample, $A$ and $B$ are proper subsets of $X.$ I think you meant to say "when $A\cup B$ is a proper subset of $X.$" $\endgroup$ – bof Sep 26 '17 at 12:31
  • $\begingroup$ Ah yes. Correct. :) $\endgroup$ – gune Sep 27 '17 at 13:26
  • $\begingroup$ Hi Charith, so do you mean my conclusion that $A^{c} \cap B^{c} \ne \emptyset$ is correct? so my conclusion is fine based on my guess from the Venn diagram. For example, the Fifth Diagram here: cs.uni.edu/~campbell/stat/venn.html , the intersection is clearly non-empty. $\endgroup$ – john_w Sep 28 '17 at 4:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.