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Unordered Triples
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Characteristic Functions
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Set Powers


We want to consider all the subsets of a given set.


We do not yet have a principle stating that such a set exists, but our intuition suggests that it does.

For every set, there exists a set of its subsets.

We call the existence of this set the principle of powers and we call the set the power set.1 As usual, the principle of extension gives uniqueness (see Set Equality). The power set of a set includes the set itself and the empty set.


Let $A$ denote a set. We denote the power set of $A$ by $\powerset{A}$, read aloud as “powerset of A.” $A \in \powerset{A}$ and $\varnothing \in \powerset{A}$. However, $A \subset \powerset{A}$ is false.


Let $a, b, c$ denote distinct objects. Let $A = \set{a, b ,c}$ and $B = \set{a, b}$. Then $B \subset A$. In other notation, $B \in \powerset{A}$. Showing each of the following is straightforward.

  1. The empty set: $\powerset{\varnothing} = \set{\varnothing}$
  2. Singletons: $\powerset{\set{a}} = \set{\varnothing, \set{a}}$
  3. Pairs: $\powerset{\set{a, b}} = \set{\varnothing, \set{a}, \set{b}, \set{a, b}}$
  4. Triples:

    \[ \powerset{\set{a, b, c}} = \{ \varnothing, \set{a}, \set{b}, \set{c}, \set{a, b}, \set{b, c}, \set{a, c}, \set{a, b, c} \} \]


We can guess the following easy properties.2

$\varnothing \in \powerset{A}$
$A \in \powerset{A}$

We call $A$ and $\varnothing$ the improper subsets of $A$. All other subsets we call proper.

Basic fact

$E \subset F \implies \powerset{E} \subset \powerset{F}$

  1. This terminology is standard, but unfortunate. Future editions may change these terms. ↩︎
  2. Future editions will expand this account. ↩︎
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