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Let $A, B,$ and $C$ be arbitrary sets taken from the positive integers.

I have to prove or disprove that: $ \text{If }A ∩ B ∩ C = ∅, \text{then } (A ⊆ \sim B) \text{ or } (A ⊆ \sim C)$

Here is my disproof using a counterexample:

If $A = \{ \}$ the empty set, $B = \{2, 3\}$, $C = \{4, 5\}$.

With these sets defined for $A, B,$ and $C$, the intersection includes the disjoint set, and then that would lead to $A$ being a subset of $B$ or $A$ being a subset of $C$ which counteracts that

if $A ∩ B ∩ C = ∅$, then $(A ⊆ \sim B)$ or $(A ⊆ \sim C)$.

Is this a sufficient proof?

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    lol thanks. now i know2011-08-12

2 Answers 2

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Let $A=\{2,3\}$, $B=\{1,3\}$, and $C=\{1,2\}$.

The intersection of the three sets is empty. But none of them is a subset of the complement of another.

By symmetry, it is sufficient for example to show that $A$ is not a subset of the complement $B^c$ of $B$.

Note that $B^c$ consists of all integers except $1$ and $3$. Since $A$ does contain $3$, $A$ is not a subset of $B^c$.

Comment: As has been pointed out in a comment by @Joe, the empty set is a subset of every set, so setting $A=\emptyset$ cannot give you a counterexample, whatever be the choice of $B$ and $C$.

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    @Krysten: More than that. If you permute $A$, $B$, and $C$ in any way, the basic pattern remains unchanged. Thus we get two bonuses: (i) if $X$ and $Y$ are two distinct sets chosen from $A$, $B$, $C$ then $X$ is not a subset of $Y^c$ and (ii) checking the property for one pair effectively checks it for all $6$ possible ordered pairs $(X,Y)$.2011-08-12
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As another counterexample, let $A=\mathbb{N}$, and $B$ and $C$ be disjoint sets with at least one element each.

Elaborating: On the one hand, since $B$ and $C$ are disjoint, $ A \cap B \cap C = \mathbb{N} \cap B \cap C = B \cap C = \emptyset. $ On the other hand, $A = \mathbb{N}$ is not contained in the complement of $B$, since $B$ is not empty, and the same for $C$.