Direct or Cartesian product of two sets

A direct or Cartesian product of two sets is a set whose elements are all sorted ordered pairs of elements of the original sets.

The concept of a direct product is naturally generalized to the product of sets with an additional structure (algebraic, topological, etc.) since the product of sets often inherits the structures found on the original sets.

Direct product in set theory [edit]

The product of two sets [edit]

Let two sets be given and . Direct product set and sets there are so many whose elements are ordered pairs for all kinds and .

Displays of the product of sets in its factors - and - called coordinate functions .

The product of a finite family of sets is defined similarly.

Comments [edit]

Strictly speaking, associative identity does not take place, but by the existence of a natural one-to-one correspondence between sets and this distinction can often be neglected.

Cartesian degree [edit]

-th Cartesian degree set determined for whole non-negative , as -fold Cartesian product to myself:

Usually referred to as or .

With positive Cartesian degree consists of all ordered sets (tuples) of elements from lengths . So real space (a set of tuples of three real numbers), there are 3 degree of the set of real numbers

With Cartesian degree by definition, contains a single element — an empty tuple.

Direct product of a family of sets [edit]

In general, for an arbitrary family of sets (not necessarily different) (the set of indices can be infinite) direct product defined as a set of functions that associate each element set element :

Mappings called projections .

In particular, for a finite family of sets any function with the condition equivalent to some tuples composed of set elements so that on the second place of the tuple is an element of the set . Therefore, the Cartesian (direct) product of a finite number of sets can be written like this:

Projections are defined as follows:

Direct Product Mappings [edit]

Let be - mapping from at , but - mapping from at . Their direct product called mapping from at : .

Similar to the above, this definition is generalized to multiple and infinite products.

Impact on mathematical structures [edit]

Direct product of groups [edit]

Direct (Cartesian) product of two groups and Is a group of all pairs of elements with the operation of component multiplication: . This group is referred to as . Associativity of multiplication operations in a group follows from the associativity of operations of multiplied groups. Multipliers and are isomorphic to two normal subgroups of their work, and respectively. The intersection of these subgroups consists of one element. which is a unit of a work group. Coordinate functions of the product of groups are homomorphisms.

This definition applies to an arbitrary number of groups multiplied. In the case of a finite number, the direct product is isomorphic to a direct sum. The difference occurs with an infinite number of factors.

In general, where and . (The operation on the right side is a group operation .) The unit of the work group will be a sequence composed of the units of all the groups being multiplied: . For example, for a countable number of groups: where the right side is the set of all infinite binary sequences.

Subgroup on the set of all whose carrier (i.e. ) is finite, called the direct sum . For example, the direct sum of the same set of sets contains all binary sequences with a finite number of ones, and they can be interpreted as binary representations of natural numbers.

Direct product of other algebraic structures [edit]

Similar to the product of groups, it is possible to define the products of rings, algebras, modules and linear spaces, and in the definition of the direct product (see above) should be replaced with zero. The definition of a product of two (or a finite number) of objects coincides with the definition of a direct sum . However, generally speaking, the direct sum differs from the direct product: for example, the direct product of a countable set of copies are the space of all sequences of real numbers, whereas the direct sum is the space of those sequences that have only a finite number of non-zero members (so-called finite sequences ).

Direct product of topological spaces [edit]

Let be and - two topological spaces. Work Topology given by the base, consisting of all sorts of works where —Open subset and - open subset .

The definition is easily generalized to the case of the product of several spaces. For infinite work the definition is complicated. Define an open cylinder where and - open subset .

The topology of an infinite product will be defined by a base made up of all possible intersections of a finite number of open cylinders (this topology is similar to the compact-open topology of mapping spaces if we take the index set having a discrete topology).

The Tikhonov theorem asserts the compactness of products of any number of compact spaces; however, for infinite products, it cannot be proved without using the axiom of choice (or the assertions of the theory of sets equivalent to it).

Also, Aleksandrov's theorem shows that any topological space can be embedded in the (infinite) product of connected colons, if only the Kolmogorov axiom is fulfilled.

Direct product of graphs [edit]

The set of vertices of the direct product of two graphs and is defined as the product of the vertices of the factor graphs. The edges of the vertices will be connected to the following pairs of vertices:

• where and - connected by the edge of the top of the graph , but - arbitrary vertex of the graph ;
• where - arbitrary vertex of the graph , but and - connected by the edge of the top of the graph .

In other words, the set of edges of a graph product is the union of two products: the edges of the first to the vertices of the second, and the vertices of the first to the edges of the second.

Variations and generalizations [edit]

The idea of ​​direct work was further developed in the theory of categories, where it served as the basis for the concept of the product of objects . Informally, the product of two objects and - this is the most common object in this category for which there are projections on and . In many categories (sets, groups, graphs, ...) the product of objects is precisely their direct product. It is important that in most cases it is important not so much a specific definition of a direct work, as the above property of universality. Different definitions will give isomorphic objects.

See also [edit]

• Disjoint union
• Semi-straight product
• Direct amount
• Tensor product
• Cartesian coordinates
• Set operations
• Combinatorics

// C #
using System;
using System.Collections.Generic;
using System.Text;

namespace mz
{
class program
{
static void Main (string [] args)
{
Console.WriteLine ("Enter the number of elements in the first and second sets, respectively:");
int xcount = Convert.ToInt32 (Console.ReadLine ());
int ycount = Convert.ToInt32 (Console.ReadLine ());
string [] X = new string [xcount];
string [] Y = new string [ycount];
Console.WriteLine ("Enter the elements of the first set:");
for (int i = 0; i <xcount; i ++)
X = Console.ReadLine ();
Console.WriteLine ("Enter the elements of the second set:");
for (int i = 0; i <ycount; i ++)
Y = Console.ReadLine ();
XY xy = new XY ();
string [] s = xy.MulXY (X, Y);
Console.WriteLine ("Result:");
xy.PrintMulXY (s);
Console.ReadLine ();
}
}
public class XY
{
public string [] MulXY (string [] X, string [] Y)
{
string [] res = new string [X.Length * Y.Length];
int k = 0;
foreach (string i in X)
foreach (string j in Y)
{
res [k] = "(" + i + "," + j + ")";
k ++;
}
return res;
}
public void PrintMulXY (string [] strMulXY)
{
foreach (string s in strMulXY)
Console.WriteLine ("{0}", s);
return;
}
}
}

In C ++, using STL.