# Cartesian Product of Family/Examples

## Examples of Cartesian Products of Families

### Household Pets

Informally it is like this.

We have a set of sets which are indexed by the indexing set can be anything, as long as it is ordered.

We think of it as numbers, but it could be anything.

The indexing set indentifies both sets, and elements in the tuple.

Let $I$ be the set of household pets:

$I := \set {\text {Bruiser}, \text {Claude} }$

Let $T$ be the set of toys they play with:

$T:= \set {\text {rubber bone}, \text {chewed tennis ball}, \text {missing slipper}, \text {jingly roller}, \text {fluffy mouse on a string}, \text {table tennis ball} }$

The set of toys played with by $\text {Bruiser}$ consist of:

$\set {\text {rubber bone}, \text {chewed tennis ball}, \text {missing slipper} }$

The set of toys played with by $\text {Claude}$ consist of:

$\set {\text {table tennis ball}, \text {jingly roller}, \text {fluffy mouse on a string} }$

Thus we have a tuple of sets:

$\tuple {\set {\text {rubber bone}, \text {chewed tennis ball}, \text {missing slipper} }, \set {\text {table tennis ball}, \text {jingly roller}, \text {fluffy mouse on a string} } }$

Let $I = \set {\text {Bruiser}, \text {Claude} }$ be considered as an indexing set.

The tuple indexed by $\text {Bruiser}$ is:

$\set {\text {rubber bone}, \text {chewed tennis ball}, \text {missing slipper} }$

The tuple indexed by $\text {Claude}$ is:

$\set {\text {table tennis ball}, \text {jingly roller}, \text {fluffy mouse on a string} }$

The Cartesian product of the toys of $\text {Bruiser}$ and $\text {Claude}$ is:

$\left \{ {\tuple {\text {rubber bone}, \text {jingly roller} }, \tuple {\text {chewed tennis ball}, \text {jingly roller} }, \tuple {\text {missing slipper}, \text {jingly roller} }, }\right.$
$\left. {\tuple {\text {rubber bone}, \text {table tennis ball} }, \tuple {\text {chewed tennis ball}, \text {table tennis ball} }, \tuple {\text {missing slipper}, \text {table tennis ball} }, }\right.$
$\left. {\tuple {\text {rubber bone}, \text {fluffy mouse on a string} }, \tuple {\text {chewed tennis ball}, \text {fluffy mouse on a string} }, \tuple {\text {missing slipper}, \text {fluffy mouse on a string} } }\right\}$

Each of these tuples is indexed by $\set {\text {Bruiser}, \text {Claude} }$.

So $\tuple {\text {chewed tennis ball}, \text {fluffy mouse on a string} }$ indexed by $\text {Bruiser}$ is $\text {chewed tennis ball}$.

$\tuple {\text {chewed tennis ball}, \text {fluffy mouse on a string} }$ indexed by $\text {Claude}$ is $\text {fluffy mouse on a string}$.

### $1$ and $2$: Von Neumann Construction

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:

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