# Empty Intersection iff Subset of Complement/Corollary

## Corollary to Empty Intersection iff Subset of Complement

Let $A, B, S$ be sets such that $A, B \subseteq S$.

Then:

$\exists X \in \powerset S: \paren {A \cap X} \cup \paren {B \cap \complement_S \paren X} = \O \iff A \cap B = \O$

where $\overline X$ denotes the relative complement of $X$ in $S$.

## Proof 1

Let there exist such a set $X$.

Then:

 $\displaystyle$  $\displaystyle \paren {A \cap X} \cup \paren {B \cap \complement_S \paren X} = \O$ $\displaystyle$ $\leadstoandfrom$ $\displaystyle A \cap X = \O \land B \cap \complement_S \paren X = \O$ Union is Empty iff Sets are Empty $\displaystyle$ $\leadstoandfrom$ $\displaystyle A \subseteq \complement_S \paren X \land B \subseteq X$ Empty Intersection iff Subset of Complement $\displaystyle$ $\leadstoandfrom$ $\displaystyle A \cap B = \O$

$\blacksquare$

## Proof 2

We have:

result $\paren {A \cap C} \cup \paren {B \cap \map \complement C} = \O \iff B \subseteq C \subseteq \map \complement A$:

where the universe $\Bbb U$ is posited.

Let $S$ take the position of $\Bbb U$.

Let $C = X$.

Then we have:

$\paren {A \cap X} \cup \paren {B \cap \relcomp S X} = \O \iff B \subseteq X \subseteq \relcomp S A$

Thus we have shown that:

$B \subseteq \relcomp S A$

and it follows from Empty Intersection iff Subset of Complement that:

$A \cap B = \O$

$\blacksquare$