Cartesian Product of Two Sets

Let A and B be two sets, Cartesian productA × B is the set of all ordered pair of elements from A and B
A × B = {{x, y} : x ∈ A, y ∈ B}

Let A = {a, b, c} and B = {d, e, f}
The Cartesian product of two sets is
A x B = {a, d}, {a, e}, {a, f}, {b, d}, {b, e}, {b, f}, {c, d}, {c, e}, {c, f}}

A has 3 elements and B also has 3 elements. The Cartesian Product has 3 x 3 = 9 elements.



In general, if there are m elements in set A and n elements in B, the number of elements in the Cartesian Product is m x n


Given two finite non-empty sets, write a program to print Cartesian Product.

Examples :

Input : A = {1, 2}, B = {3, 4}
Output : A × B = {{1, 3}, {1, 4}, {2, 3}, {2, 4}}

Input  : A = {1, 2, 3} B = {4, 5, 6}
Output : A × B = {{1, 4}, {1, 5}, {1, 6}, {2, 4}, 
         {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}}

CPP

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// C++ Program to find the Cartesian Product of Two Sets
#include <stdio.h>
  
void findCart(int arr1[], int arr2[], int n, int n1)
{
    for (int i = 0; i < n; i++)
        for (int j = 0; j < n1; j++)
            printf("{%d, %d}, ", arr1[i], arr2[j]);
}
  
int main()
{
    int arr1[] = { 1, 2, 3 }; // first set
    int arr2[] = { 4, 5, 6 }; // second set
    int n1 = sizeof(arr1) / sizeof(arr1[0]);
    int n2 = sizeof(arr2) / sizeof(arr2[0]);
    findCart(arr1, arr2, n1, n2);
    return 0;
}

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Java

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// Java Program to find the 
// Cartesian Product of Two Sets
import java.io.*;
import java.util.*;
  
class GFG {
  
    static void findCart(int arr1[], int arr2[], 
                                    int n, int n1)
    {
        for (int i = 0; i < n; i++)
          for (int j = 0; j < n1; j++)
            System.out.print("{"+ arr1[i]+", "
                             + arr2[j]+"}, ");
    }
    // Driver code
    public static void main (String[] args) {
          
        // first set
        int arr1[] = { 1, 2, 3 }; 
          
        // second set
        int arr2[] = { 4, 5, 6 }; 
          
        int n1 = arr1.length;
        int n2 = arr2.length;
        findCart(arr1, arr2, n1, n2);
    }
}
  
  
// This code is contributed by Nikita Tiwari.

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Python3

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# Python3 Program to find the
# Cartesian Product of Two Sets
  
def findCart(arr1, arr2, n, n1):
  
    for i in range(0,n):
        for j in range(0,n1):
            print("{",arr1[i],", ",arr2[j],"}, ",sep="",end="")
  
# Driver code
arr1 = [ 1, 2, 3 ] # first set
arr2 = [ 4, 5, 6 ] # second set
  
n1 = len(arr1) # sizeof(arr1[0])
n2 = len(arr2) # sizeof(arr2[0]);
  
findCart(arr1, arr2, n1, n2);
  
# This code is contributed
# by Smitha Dinesh Semwal

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C#

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// C# Program to find the 
// Cartesian Product of Two Sets
using System;
  
class GFG {
  
    static void findCart(int []arr1, int []arr2, 
                                    int n, int n1)
    {
        for (int i = 0; i < n; i++)
            for (int j = 0; j < n1; j++)
                Console.Write("{" + arr1[i] + ", "
                                + arr2[j] + "}, ");
    }
      
    // Driver code
    public static void Main () {
          
        // first set
        int []arr1 = { 1, 2, 3 }; 
          
        // second set
        int []arr2 = { 4, 5, 6 }; 
          
        int n1 = arr1.Length;
        int n2 = arr2.Length;
          
        findCart(arr1, arr2, n1, n2);
    }
}
  
  
// This code is contributed by vt_m.

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PHP

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<?php
// PHP Program to find the
// Cartesian Product of Two Sets
  
function findCart($arr1, $arr2, $n, $n1)
{
    for ($i = 0; $i < $n; $i++)
        for ( $j = 0; $j < $n1; $j++)
            echo "{", $arr1[$i] ," , "
                      $arr2[$j], "}",",";
}
  
// Driver Code
  
// first set
$arr1 = array ( 1, 2, 3 );
  
// second set
$arr2 = array ( 4, 5, 6 ); 
$n1 = sizeof($arr1) ;
$n2 = sizeof($arr2);
findCart($arr1, $arr2, $n1, $n2);
  
// This code is contributed by m_kit.
?>

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Output :

{{1, 4}, {1, 5}, {1, 6}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}}

Practical Examples:
1) A set of playing cards is Cartesian product of a four element set to a set of 13 elements.

2) A two dimensional coordinate system is a Cartesian product of two sets of real numbers.

Reference:
https://en.wikipedia.org/wiki/Cartesian_product



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