| Version 5 |
Version 4 |
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Given any family of sets $\{A_j\}_{j \in J}$ indexed by an index set $J$, the \emph{generalized cartesian product}
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Given any family of sets $\{A_j\}_{j \in J}$ indexed by an index set $J$, the {\em generalized cartesian product}
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| \prod_{j \in J} A_j |
\prod_{j \in J} A_j |
| is the set of all functions |
is the set of all functions |
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f\colon J \to \bigcup_{j \in J} A_j
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f: J \to \bigcup_{j \in J} A_j
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| such that $f(j) \in A_j$ for all $j \in J$. |
such that $f(j) \in A_j$ for all $j \in J$. |
| For each $i \in J$, the \emph{projection map} |
For each $i \in J$, the {\em projection map} $$\pi_i: \prod_{j \in J} A_j \to A_i$$ is the function defined by |
| \pi_i\colon \prod_{j \in J} A_j \to A_i |
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| is the function defined by |
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| \pi_i(f) := f(i). |
\pi_i(f) := f(i). |
| The axiom of choice is the statement that the generalized Cartesian product of nonempyy sets is nonempty; this has surprising and counterintuitive consequences. |
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