Indexed Cartesian Space/Examples/Singleton Index
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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 $1.6.5$