Power Set/Examples

Examples of Power Sets

Set of 3 Elements

Let $S = \set {a, b, c}$.

Then the power set of $S$ is:

$\powerset S = \set {\O, \set a, \set b, \set c, \set {a, b}, \set {b, c}, \set {a, c}, S}$

and so has $2^3 = 8$ elements.

Note that while $\set a \in \powerset S$, $a \notin \powerset S$.

Arbitrary Example $1$

Let $A = \set {1, 2}$.

Then:

$\powerset A = \set {\O, \set 1, \set 2, \set {1, 2} }$

Empty Set

Let $\O$ denote the empty set.

Then the power set of $\O$ is:

$\powerset \O = \set \O$

and so has $2^0 = 1$ element.

Set containing Empty Set

Let $\O$ denote the empty set.

Let $S$ be the set defined as:

$S = \set \O$

Then the power set of $S$ is:

$\powerset S = \set {\O, \set \O}$

and so has $2^1 = 2$ elements.

Axiomatic Definition of 2

Let $\O$ denote the empty set.

Let $S$ be the set defined as the $2$nd element of the von Neumann construction of the natural numbers:

$S = \set {\O, \set \O}$

Then the power set of $S$ is:

$\powerset S = \set {\O, \set \O, \set {\set \O}, \set {\O, \set \O} }$

and so has $2^2 = 4$ elements.

Nested Sets of Empty Sets

Let $\O$ denote the empty set.

Let $S$ be the set defined as:

$S = \set {\O, \set \O, \set {\set \O} }$

Then the power set of $S$ is:

$\powerset S = \set {\O, \set \O, \set {\set \O}, \set {\set {\set \O} }, \set {\O, \set \O}, \set {\O, \set {\set \O} }, \set {\set \O, \set {\set \O} }, \set {\O, \set \O, \set {\set \O} } }$

and so has $2^3 = 8$ elements.