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

$\blacksquare$

Sources

• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions: Exercise $1.6.5$