Set Intersection/Set of Sets/Examples

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.