Set of Sets/Examples

Examples of Sets of Sets

Set of Arbitrary Sets: $1$

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:

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

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

Set of Arbitrary Sets: $2$

Let $A$ be the set of (strictly) positive odd integers less than $8$.

Let $B$ be the set of (strictly) positive even integers less than $8$.

Then:

\(\ds A\)

\(=\)

\(\ds \set {1, 3, 5, 7}\)

\(\ds B\)

\(=\)

\(\ds \set {2, 4, 6}\)

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

Then:

$\mathscr S = \set {\set {1, 3, 5, 7}, \set {2, 4, 6} }$

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 $\Z$.