Product set/Arithmetic operations/Geometric examples/Introduction/Section

We want to describe within set theory the arithmetic operations on the number sets like addition and multiplication. For the addition (say on ${}\mathbb {N}$), two natural numbers ${}a$ and ${}b$ are added to yield another natural number, namely ${}a+b$. The set of pairs constitute the product set and the adding is interpreted as a mapping on the product set.

We define.

Definition

Suppose that two sets ${}L$ and ${}M$ are given. Then the set

${}L\times M={\left\{(x,y)\mid x\in L,\,y\in M\right\}}\,$

is called the product set of the sets.

The elements of a product set are called pairs and denoted by ${}(x,y)$. Here the ordering is essential. The product set consists of all pair combinations, where in the first component there is an element of the first set and in the second component there is an element of the second set. Two pairs are equal if and only if they are equal in both components.

If one of the sets is empty, then so is the product set. If both sets are finite, say the first with ${}n$ elements and the second with ${}k$ elements, then the product set has ${}n\cdot k$ elements. It is also possible to form the product set of more than two sets.

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 deduce easily 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.

For a product set it is also possible that both sets are equal. Then one has to be careful not to confuse the components.

Example

A chess board (meaning the set of squares of a chess board where a chess piece may stand) is the product set ${}\{a,b,c,d,e,f,g,h\}\times \{1,2,3,4,5,6,7,8\}$. Every square is a pair, e.g. ${}(a,1),(d,4),(c,7)$. Because the two component sets are different, one may write instead of pair notation simply ${}a1,d4,c7$. This notation is the starting point to describe chess positions, and complete chess games.

The product set ${}\mathbb {R} \times \mathbb {R}$ is thought of as a plane, one denotes it also by ${}\mathbb {R} ^{2}$. The product set ${}\mathbb {Z} \times \mathbb {Z}$ is a set of lattice points.

Example

The cylinder (its surface) is the product set of a circle and a line segment.

Let ${}S$ denote a circle (the circumference), and let ${}I$ be a line segment. The circle is a subset of a plane ${}E$, and the line segment is a subset of a line ${}G$, so that for the product sets, we have the relation

${}S\times I\subseteq E\times G\,.$

The product set ${}E\times G$ is the three-dimensional space, and the product set ${}S\times I$ is the surface of a cylinder.