Indexed Cartesian Space/Examples/Singleton Index

From ProofWiki
Jump to navigation Jump to search

Example of Indexed Cartesian Space

Let $X$ be a set.

Then the indexed Cartesian space $X^{\set 1}$ is defined as:

$X^{\set 1} = \set 1 \times X$


\(\ds X^{\set 1}\) \(=\) \(\ds \set {f: \paren {f: \set 1 \to X} \land \paren {\forall i \in \set 1: \paren {\map f i \in X} } }\) Definition 2 of Indexed Cartesian Space
\(\ds \) \(=\) \(\ds \set {\tuple {i, s}: i \in \set 1, s \in X}\) Definition of Mapping
\(\ds \) \(=\) \(\ds \set {\tuple {1, s}: s \in X}\) Definition of Singleton
\(\ds \) \(=\) \(\ds \set 1 \times X\) Definition of Cartesian Product