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

  • $\begingroup$ 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

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

  • 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

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