Set of Sets/Examples

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Examples of Sets of Sets

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:

$\mathscr S = \set {\set {1, 2, 3, 4}, \set {a, 3, 4}, \set {2, a} }$


Note that none of $a, 1, 2, 3, 4$ are elements of $S$.


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:

$\mathscr S := \set {\set 1, \set {1, 2}, \set {1, 2, 3}, \ldots}$

and we have that:

$\mathscr S \subsetneq \powerset \Z$

where $\powerset \Z$ denotes the power set of $\Z$.