De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection/Proof
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Theorem
- $\ds S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$
where:
- $\ds \bigcap \mathbb T := \set {x: \forall T' \in \mathbb T: x \in T'}$
that is, the intersection of $\mathbb T$
Proof
Suppose:
- $\ds x \in S \setminus \bigcap \mathbb T$
Note that by Set Difference is Subset we have that $x \in S$ (we need this later).
Then:
\(\ds x\) | \(\in\) | \(\ds S \setminus \bigcap \mathbb T\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\notin\) | \(\ds \bigcap \mathbb T\) | Definition of Set Difference | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \neg \leftparen {\forall T' \in \mathbb T}: \, \) | \(\ds x\) | \(\in\) | \(\ds \rightparen {T'}\) | Definition of Intersection of Set of Sets | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists T' \in \mathbb T: \, \) | \(\ds x\) | \(\notin\) | \(\ds T'\) | Denial of Universality | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists T' \in \mathbb T: \, \) | \(\ds x\) | \(\in\) | \(\ds S \setminus T'\) | Definition of Set Difference: note $x \in S$ from above | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}\) | Definition of Union of Set of Sets |
Therefore:
- $\ds S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$
$\blacksquare$
Source of Name
This entry was named for Augustus De Morgan.
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text I$: Sets and Functions: Exercise $\text{D}$