Definition:Set Intersection/Set of Sets

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Definition

Let $\Bbb S$ be a set of sets

The intersection of $\Bbb S$ is:

$\ds \bigcap \Bbb S := \set {x: \forall S \in \Bbb S: x \in S}$

That is, the set of all objects that are elements of all the elements of $\Bbb S$.


Thus:

$\ds \bigcap \set {S, T} := S \cap T$


Also denoted as

Some sources denote $\ds \bigcap \mathbb S$ as $\ds \bigcap_{S \mathop \in \mathbb S} S$.


Examples

Set of Arbitrary Sets

Let:

\(\displaystyle A\) \(=\) \(\displaystyle \set {1, 2, 3, 4}\)
\(\displaystyle B\) \(=\) \(\displaystyle \set {a, 3, 4}\)
\(\displaystyle C\) \(=\) \(\displaystyle \set {2, a}\)


Let $\mathscr S = \set {A, B, C}$.

Then:

$\displaystyle \bigcap \mathscr S = \O$


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:

$\displaystyle \bigcap \mathscr S = \set 1$


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:

$\displaystyle \bigcap \SS = \O$


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:

$\displaystyle \bigcap \TT$ is not empty.


Also see

  • Results about set intersection can be found here.


Sources