Set Intersection/Set of Sets/Examples
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Examples of Intersections of Sets of Sets
Set of Arbitrary Sets
Let:
\(\ds A\) | \(=\) | \(\ds \set {1, 2, 3, 4}\) | ||||||||||||
\(\ds B\) | \(=\) | \(\ds \set {a, 3, 4}\) | ||||||||||||
\(\ds C\) | \(=\) | \(\ds \set {2, a}\) |
Let $\mathscr S = \set {A, B, C}$.
Then:
- $\bigcap \mathscr S = \O$
Set of Initial Segments
Let $\Z$ denote the set of integers.
Let $\map \Z n$ denote the initial segment of $\Z_{> 0}$:
- $\map \Z n = \set {1, 2, \ldots, n}$
Let $\mathscr S := \set {\map \Z n: n \in \Z_{> 0} }$
That is, $\mathscr S$ is the set of all initial segments of $\Z_{> 0}$.
Then:
- $\bigcap \mathscr S = \set 1$
Set of Unbounded Above Open Real Intervals
Let $\R$ denote the set of real numbers.
For a given $a \in \R$, let $S_a$ denote the (real) interval:
- $S_a = \openint a \to = \set {x \in \R: x > a}$
Let $\SS$ denote the family of sets indexed by $\R$:
- $\SS := \family {S_a}_{a \mathop \in \R}$
Then:
- $\bigcap \SS = \O$
Finite Subfamily of Unbounded Above Open Real Intervals
Let $\R$ denote the set of real numbers.
For a given $a \in \R$, let $S_a$ denote the (real) interval:
- $S_a = \openint a \to = \set {x \in \R: x > a}$
Let $\SS$ denote the family of sets indexed by $\R$:
- $\SS := \family {S_a}_{a \mathop \in \R}$
Let $\TT$ be a finite subfamily of $\SS$.
Then:
- $\bigcap \TT$ is not empty.