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

From Wikiversity
Jump to navigation Jump to search

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

We define.[1]

Suppose that two sets and are given. Then the set

is called the product set of the sets.

The elements of a product set are called pairs and denoted by . 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 elements and the second with elements, then the product set has elements. It is also possible to form the product set of more than two sets.

Let be the set of all first names, and be the set of all last names. Then

is the set of all names. The elements of this set are in pair notation , and . 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.

A chess board (meaning the set of squares of a chess board where a chess piece may stand) is the product set . Every square is a pair, e.g. . Because the two component sets are different, one may write instead of pair notation simply . This notation is the starting point to describe chess positions, and complete chess games.

The product set is thought of as a plane, one denotes it also by . The product set is a set of lattice points.

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

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

The product set is the three-dimensional space, and the product set is the surface of a cylinder.

  1. In mathematics, definitions are usually presented as such and get a number so that it is easy to refer to them. The definition contains the description of a situation where only concepts are used which have been defined before. In this situation, a new concept together with a name for it is introduced. This name is printed in a certain font, typically in italic. The new concept can be formulated without the new name, the new name is an abbreviation for the new concept. Quite often, the concepts depend on parameters, like the product set depends on its component sets. The names are often chosen arbitrarily, the meaning of the word within the mathematical context can be understood only via the explicit definition and not via its meaning in everyday life.