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Direct or Cartesian product of two sets

Lecture



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.

Content

  • 1 Direct product in set theory
    • 1.1 The product of two sets
    • 1.2 Comments
    • 1.3 Cartesian degree
    • 1.4 The direct product of a family of sets
  • 2 Direct product of mappings
  • 3 Impact on mathematical structures
    • 3.1 Direct product of groups
    • 3.2 Direct product of other algebraic structures
    • 3.3 Direct product of topological spaces
    • 3.4 Direct product of graphs
  • 4 Variations and generalizations
  • 5 See also

Direct product in set theory [edit]

The product of two sets [edit]

at at at at at at at at
and and and and and and and and
to to to to to to to to
The product of the set {in, and, k}
many rainbow colors

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

Displays of the product of sets in its factors -   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets - called coordinate functions .

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

Comments [edit]

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

Cartesian degree [edit]

000 001 002 010 011 012 020 021 022
100 101 102 110 111 112 120 121 122
200 201 202 210 211 212 220 221 222
{0, 1, 2} 3 , 3 3 = 27 elements

  Direct or Cartesian product of two sets -th Cartesian degree set   Direct or Cartesian product of two sets determined for whole non-negative   Direct or Cartesian product of two sets , as   Direct or Cartesian product of two sets -fold Cartesian product   Direct or Cartesian product of two sets to myself:

  Direct or Cartesian product of two sets

Usually referred to as   Direct or Cartesian product of two sets or   Direct or Cartesian product of two sets .

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

With   Direct or Cartesian product of two sets Cartesian degree   Direct or Cartesian product of two sets 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)   Direct or Cartesian product of two sets (the set of indices can be infinite) direct product   Direct or Cartesian product of two sets defined as a set of functions that associate each element   Direct or Cartesian product of two sets set element   Direct or Cartesian product of two sets :

  Direct or Cartesian product of two sets

Mappings   Direct or Cartesian product of two sets called projections .

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

  Direct or Cartesian product of two sets

Projections are defined as follows:   Direct or Cartesian product of two sets

Direct Product Mappings [edit]

Let be   Direct or Cartesian product of two sets - mapping from   Direct or Cartesian product of two sets at   Direct or Cartesian product of two sets , but   Direct or Cartesian product of two sets - mapping from   Direct or Cartesian product of two sets at   Direct or Cartesian product of two sets . Their direct product   Direct or Cartesian product of two sets called mapping from   Direct or Cartesian product of two sets at   Direct or Cartesian product of two sets :   Direct or Cartesian product of two sets .

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   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets Is a group of all pairs of elements   Direct or Cartesian product of two sets with the operation of component multiplication:   Direct or Cartesian product of two sets . This group is referred to as   Direct or Cartesian product of two sets . Associativity of multiplication operations in a group   Direct or Cartesian product of two sets follows from the associativity of operations of multiplied groups. Multipliers   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets are isomorphic to two normal subgroups of their work,   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets respectively. The intersection of these subgroups consists of one element.   Direct or Cartesian product of two sets 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,   Direct or Cartesian product of two sets where   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets . (The operation on the right side is a group operation   Direct or Cartesian product of two sets .) The unit of the work group will be a sequence composed of the units of all the groups being multiplied:   Direct or Cartesian product of two sets . For example, for a countable number of groups:   Direct or Cartesian product of two sets where the right side is the set of all infinite binary sequences.

Subgroup on the set of all   Direct or Cartesian product of two sets whose carrier (i.e.   Direct or Cartesian product of two sets ) is finite, called the direct sum . For example, the direct sum of the same set of sets   Direct or Cartesian product of two 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   Direct or Cartesian product of two sets (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   Direct or Cartesian product of two sets 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   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets - two topological spaces. Work Topology   Direct or Cartesian product of two sets given by the base, consisting of all sorts of works   Direct or Cartesian product of two sets where   Direct or Cartesian product of two sets —Open subset   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets - open subset   Direct or Cartesian product of two sets .

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

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   Direct or Cartesian product of two sets 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]

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The set of vertices of the direct product of two graphs   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets 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:

  •   Direct or Cartesian product of two sets where   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets - connected by the edge of the top of the graph   Direct or Cartesian product of two sets , but   Direct or Cartesian product of two sets - arbitrary vertex of the graph   Direct or Cartesian product of two sets ;
  •   Direct or Cartesian product of two sets where   Direct or Cartesian product of two sets - arbitrary vertex of the graph   Direct or Cartesian product of two sets , but   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets - connected by the edge of the top of the graph   Direct or Cartesian product of two sets .

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   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets - this is the most common object in this category for which there are projections on   Direct or Cartesian product of two sets and   Direct or Cartesian product of two sets . 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.

Code:
#include "stdafx.h"
#include <iostream>
#include <vector>
#include <utility>
#include <string>
using namespace std;
int _tmain (int argc, _TCHAR * argv [])
{
wcout.imbue (locale (". 866"));
vector <string> coll1;
vector <string> coll2;
int size1, size2;
wcout << L "Enter the size of the sets" << endl;
cin >> size1;
cin >> size2;
string s;
wcout << L "Enter the elements of the 1st set" << endl;
for (int i = 'a'; i <'a' + size1; ++ i) {
cin >> s;
coll1.push_back (s);
}
wcout << L "Enter the elements of the 2nd set" << endl;
for (int i = 0; i <size2; ++ i) {
cin >> s;
coll2.push_back (s);
}
wcout << L "\ t1-th set" << endl;
for (int i = 0; i <coll1.size (); ++ i)
cout << coll1 << '';
cout << endl;
wcout << L "\ t2-th set" << endl;
for (int i = 0; i <coll2.size (); ++ i)
cout << coll2 << '';
cout << endl;
vector <pair <string, string>> coll3;
for (int i = 0; i <coll1.size (); ++ i) {
for (int j = 0; j <coll2.size (); ++ j)
coll3.push_back (make_pair (coll1, coll2 [j]));
}
wcout << L "\ tDescription product" << endl;
for (int i = 0; i <coll3.size (); ++ i)
cout << "<" << coll3.first << '' << coll3.second << ">"
<< "";
cout << endl;
return 0;
}

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Discrete Math. Set theory. Graph theory. Combinatorics.

Terms: Discrete Math. Set theory. Graph theory. Combinatorics.