(A cap C) cup (B cap Complement C) = Empty iff B subset C subset Complement A

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Theorem

Let $A$, $B$ and $C$ be subsets of a universe $\Bbb U$.

Then:

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

where $\map \complement C$ denotes the complement of $C$ in $\Bbb U$.


Proof

\(\displaystyle \paren {A \cap C} \cup \paren {B \cap \map \complement C}\) \(=\) \(\displaystyle \O\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle A \cap C\) \(=\) \(\displaystyle \O\) Union is Empty iff Sets are Empty
\(\, \displaystyle \land \, \) \(\displaystyle B \cap \map \complement C\) \(=\) \(\displaystyle \O\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle C\) \(\subseteq\) \(\displaystyle \map \complement A\) Empty Intersection iff Subset of Complement
\(\, \displaystyle \land \, \) \(\displaystyle B\) \(\subseteq\) \(\displaystyle C\) Intersection with Complement is Empty iff Subset
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle B\) \(\subseteq\) \(\displaystyle C \subseteq \map \complement A\)

$\blacksquare$


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