| Version current |
Version 7 |
| The \emph{axiom of power set} is an axiom of Zermelo-Fraenkel set theory which postulates that for any set $X$ there exists a set $\mathcal{P}(X)$, called the \emph{power set} of $X$, consisting of all subsets of $X$. In symbols, it reads: |
The \emph{axiom of power set} is an axiom of Zermelo-Fraenkel set theory which postulates that for any set $X$ there exists a set $\mathcal{P}(X)$, called the \emph{power set} of $X$, consisting of all subsets of $X$. In symbols, it reads: |
| \[ |
\[ |
| \forall X \exists \mathcal{P}(X) \forall u (u \in \mathcal{P}(X) \leftrightarrow u \subseteq X). |
\forall X \exists \mathcal{P}(X) \forall u (u \in \mathcal{P}(X) \leftrightarrow u \subseteq X). |
| \] |
\] |
| In the above, $u \subseteq X$ is defined as $\forall z(z \in u \rightarrow z \in X)$. By the extensionality axiom, the set $\mathcal{P}(X)$ is unique. |
In the above, $u \subseteq X$ is defined as $\forall z(z \in u \rightarrow z \in X)$. By the extensionality axiom, the set $\mathcal{P}(X)$ is unique. |
|
|
| The Power Set Axiom allows us to define the Cartesian product of two sets $X$ and $Y$: |
The Power Set Axiom allows us to define the Cartesian product of two sets $X$ and $Y$: |
| \[ |
\[ |
| X \times Y = \{ (x, y) : x \in X \land y \in Y \}. |
X \times Y = \{ (x, y) : x \in X \land y \in Y \}. |
| \] |
\] |
|
|
| The Cartesian product is a set since |
The Cartesian product is a set since |
| \[ |
\[ |
| X \times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y)). |
X \times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y)). |
| \] |
\] |
|
|
| We may define the Cartesian product of any finite collection of sets recursively: |
We may define the Cartesian product of any finite collection of sets recursively: |
| \[ |
\[ |
| X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_{n-1}) \times X_n. |
X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_{n-1}) \times X_n. |
| \] |
\] |
