# Power Set of Doubleton

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## Theorem

Let $x, y$ be distinct objects.

Then the power set of the doubleton $\set {x, y}$ is:

$\powerset {\set {x, y}} = \big \{ \O, \set x, \set y, \set {x,y} \big \}$

## Proof

By definition of a subset:

$\set x , \set y, \set {x, y} \subseteq \set{x, y}$

Let $A \subseteq \set {x, y}$:

$A \ne \set x, \set y, \set {x,y}$

From set equality:

$\set {x,y} \not \subseteq A$
either $x \notin A$ or $y \notin A$.

Without loss of generality assume that $x \notin A$.

$A \cap \set x = \O$
$A \subseteq \set{x, y} \setminus \set x$
$A \subseteq \set y$

From set equality:

$\set y \not \subseteq A$
$y \notin A$.
$A \cap \set y = \O$
$A \subseteq \set y \setminus \set y$
$A \subseteq \O$
$\O \subseteq A$

From set equality:

$A = \O$

It follows that:

$\powerset {\set {x, y}} = \big \{ \O, \set x, \set y, \set {x,y} \big \}$

$\blacksquare$