# Cartesian Product of Family/Examples/1 and 2

## 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:

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

$\blacksquare$