Product set/First and last name/Example

Let ${}F$ be the set of all first names, and ${}L$ be the set of all last names. Then

$F\times L$

is the set of all names. The elements of this set are, in pair notation, ${}({\text{Heinz}},{\text{Miller}})$, ${}({\text{Petra}},{\text{Miller}})$ and ${}({\text{Lucy}},{\text{Sonnenschein}})$. From a name, one can easily deduce the first name and the last name by looking at the first or the second component. Even if all first names and all last names do really occur, not every combination of a first name and a last name does occur. For the product set, all possible combinations are allowed.