Empty Intersection iff Subset of Complement/Corollary/Proof 1
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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
Let there exist such a set $X$.
Then:
\(\ds \) | \(\) | \(\ds \paren {A \cap X} \cup \paren {B \cap \complement_S \paren X} = \O\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds A \cap X = \O \land B \cap \complement_S \paren X = \O\) | Union is Empty iff Sets are Empty | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds A \subseteq \complement_S \paren X \land B \subseteq X\) | Empty Intersection iff Subset of Complement | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds A \cap B = \O\) |
$\blacksquare$