Cartesian Product of Family/Examples/1 and 2

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Example of Cartesian Product of Family

Let $A_\O := \set \O$ and $A_{\set \O} := \set {\O, \set \O}$.

Thus $A_\O$ and $A_{\set \O}$ are the numbers $1$ and $2$ as defined by the Von Neumann construction.


Then:

$A_\O \times A_{\set \O} = \set {\tuple {\O, \O}, \tuple {\O, \set \O} }$

while:

$\ds \prod_{i \mathop \in A_{\set \O} } A_i = \set {\set {\tuple {\O, \O}, \tuple {\set \O, \O} }, \set {\tuple {\O, \O}, \tuple {\set \O, \set \O} } }$


Proof

First we have:

\(\ds A_\O \times A_{\set \O}\) \(=\) \(\ds \set \O \times \set {\O, \set \O}\) Definition of $A_\O$ and $A_{\set \O}$
\(\ds \) \(=\) \(\ds \set {\tuple {\O, \O}, \tuple {\O, \set \O} }\) Definition of Cartesian Product


Then:

\(\ds \prod_{i \mathop \in A_{\set \O} } A_i\) \(=\) \(\ds \prod_{i \mathop \in \set {\O, \set \O} } A_i\)
\(\ds \) \(=\) \(\ds \set {f \in \paren {\bigcup_{i \mathop \in A_{\set \O} } A_i}^{A_{\set \O} }: \forall i \in A_{\set \O}: \paren {\map f i \in A_i} }\) Definition 2 of Cartesian Product of Family


The above is deconstructed as follows.

We have that:

\(\ds \bigcup_{i \mathop \in A_{\set \O} } A_i\) \(=\) \(\ds \bigcup_{i \mathop \in \set {\O, \set \O} } A_i\) Definition of $A_{\set \O}$
\(\ds \) \(=\) \(\ds \bigcup \set {A_\O, A_{\set \O} }\) Definition of Union of Family
\(\ds \) \(=\) \(\ds \bigcup \set {\set \O, \set {\O, \set \O} }\) Definition of $A_\O$ and $A_{\set \O}$
\(\ds \) \(=\) \(\ds \set \O \cup \set {\O, \set \O}\) Union of Doubleton
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \set {\O, \set \O}\) Definition of Set Union


Hence we have that:

\(\ds \paren {\bigcup_{i \mathop \in A_{\set \O} } A_i}^{A_{\set \O} }\) \(=\) \(\ds \paren {\set {\O, \set \O} }^{A_{\set \O} }\) from above
\(\ds \) \(=\) \(\ds \paren {\set {\O, \set \O} }^{\set {\O, \set \O} }\) Definition of $A_{\set \O}$
\(\text {(2)}: \quad\) \(\ds \) \(=\) \(\ds \set {\set {\tuple {\O, \O}, \tuple {\set \O, \O} }, \set {\tuple {\O, \O}, \tuple {\set \O, \set \O} }, \set {\tuple {\O, \set \O}, \tuple {\set \O, \O} }, \set {\tuple {\O, \set \O}, \tuple {\set \O, \set \O} } }\) Definition of Set of All Mappings


Note that $(2)$ above is the set of all mappings from $\set {\O, \set \O}$ to $\set {\O, \set \O}$ as follows:

Each such mapping is a set of $2$ ordered pairs of which the first coordinates are the elements of $\set {\O, \set \O}$
From Cardinality of Set of All Mappings there are $2^2 = 4$ set of $2$ such ordered pairs.


Now we have to select the elements $f$ of $\ds \paren {\bigcup_{i \mathop \in A_{\set \O} } A_i}^{A_{\set \O} }$ such that:

$\map f i \in A_i$

for all $i \in \set {\O, \set \O}$.


We have that:

\(\ds \map f \O\) \(\in\) \(\ds A_\O\)
\(\ds \leadsto \ \ \) \(\ds \map f \O\) \(\in\) \(\ds \set \O\)
\(\ds \leadsto \ \ \) \(\ds \map f \O\) \(=\) \(\ds \O\)


Then:

\(\ds \map f {\set \O}\) \(\in\) \(\ds A_{\set \O}\)
\(\ds \leadsto \ \ \) \(\ds \map f {\set \O}\) \(\in\) \(\ds \set {\O, \set \O}\)
\(\ds \leadsto \ \ \) \(\ds \map f {\set \O}\) \(=\) \(\ds \O\)
\(\, \ds \text {or} \, \) \(\ds \map f {\set \O}\) \(=\) \(\ds \set \O\)


Hence:

$\ds \prod_{i \mathop \in A_{\set \O} } A_i = \set {\set {\tuple {\O, \O}, \tuple {\set \O, \O} }, \set {\tuple {\O, \O}, \tuple {\set \O, \set \O} } }$

$\blacksquare$


Sources

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