Indexed Cartesian Space is Set of all Mappings

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Theorem

Let $I$ be an indexing set.

Let $\displaystyle \prod_{i \mathop \in I} S$ denote the cartesian space of $S$ indexed by $I$.


Then $\displaystyle \prod_{i \mathop \in I} S$ is the set of all mappings from $I$ to $S$, and hence the notation:

$S^I := \displaystyle \prod_{i \mathop \in I} S$


Proof

Recall the definition of the cartesian space of $S$ indexed by $I$:


Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be an family of sets indexed by $I$.

Let $\displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.

Let $S$ be a set such that:

$\forall i \in I: S_i = S$


Definition 1

The Cartesian space of $S$ indexed by $I$ is the set of all families $\family {s_i}_{i \mathop \in I}$ with $s_i \in S$ for each $i \in I$:

$S_I := \displaystyle \prod_I S = \set {\family {s_i}_{i \mathop \in I}: s_i \in S}$


Definition 2

The Cartesian space of $S$ indexed by $I$ is defined and denoted as:

$\displaystyle S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {\map f i \in S} } }$



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