Indexed Cartesian Space/Examples/Singleton Index

Example of Indexed Cartesian Space

Let $X$ be a set.

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

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

Proof

 $\ds X^{\set 1}$ $=$ $\ds \set {f: \paren {f: \set 1 \to X} \land \paren {\forall i \in \set 1: \paren {\map f i \in X} } }$ 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

$\blacksquare$