Sets/Constructions/Section

Most relevant sets in mathematics arise from some basic sets like finite sets or ${}\mathbb {N}$ by certain constructions. 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

(or Cartesian product) 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.

It is possible that both sets are equal, like ${}\mathbb {R} \times \mathbb {R}$, the real plane. Then one has to be careful not to confuse the 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 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.

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.

When two geometric point sets ${}A$ and ${}B$ are given, for example, as subsets of a plane ${}E$, then we can consider the product set ${}A\times B$ as a subset of ${}E\times E$. By this procedure, we get a new geometric object, which sometimes might be realized in a smaller dimension.

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.

Another important construction, to get from a set a new set, is the power set.

Definition

For a given set ${}M$, the set consisting of all subsets of ${}M$ is called the power set of ${}M$. It is denoted by

${\mathfrak {P}}\,(M).$

We have thus

${}{\mathfrak {P}}\,(M)={\left\{T\mid T{\text{ is subset of }}M\right\}}\,.$

If ${}M$ denotes the set of all people in the course, then one can think of a subset as a party (within the course), where some people go to (we identify parties with the attending people). The power set is then the set of all possible parties. If the set has ${}n$ elements, then the power set contains ${}2^{n}$ elements.