Empty Intersection iff Subset of Complement
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Corollary to Intersection with Complement is Empty iff Subset
- $S \cap T = \O \iff S \subseteq \relcomp {} T$
where:
- $S \cap T$ denotes the intersection of $S$ and $T$
- $\O$ denotes the empty set
- $\complement$ denotes set complement
- $\subseteq$ denotes subset.
Corollary
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
\(\ds S \cap T\) | \(=\) | \(\ds \O\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds S \cap \relcomp {} {\relcomp {} T}\) | \(=\) | \(\ds \O\) | Complement of Complement | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds S\) | \(\subseteq\) | \(\ds \relcomp {} T\) | Intersection with Complement is Empty iff Subset |
$\blacksquare$
Proof 2
From Intersection with Complement is Empty iff Subset
- $S \subseteq T \iff S \cap \relcomp {} T = \O$
Then we have:
\(\ds \) | \(\) | \(\ds S \nsubseteq \relcomp {} T\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \neg \paren {\forall x \in S: x \in \relcomp {} T}\) | Definition of Subset | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \exists x \in S: x \notin \relcomp {} T\) | Denial of Universality | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \exists x \in S: x \in T\) | Definition of Set Complement | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in S \cap T\) | Definition of Set Intersection | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds S \cap T \ne \O\) | Definition of Disjoint Sets |
Thus:
\(\ds \) | \(\) | \(\ds S \cap T = \O\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \forall x \in S: x \in \relcomp {} T\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds S \subseteq \relcomp {} T\) |
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $11 \ \text{(b)}$
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 3$: Set Operations: Union, Intersection and Complement: Exercise $1 \ \text{(c)}$