Given a number n, find out the number of Reflexive Relation on a set of first n natural numbers {1, 2, ..n}.
Examples :
Input: n = 2
Output: 4
The given set A = {1, 2}. The following are reflexive relations on A * A :
{{1, 1), (2, 2)}
{(1, 1), (2, 2), (1, 2)}
{(1, 1), (2, 2), (1, 2), (2, 1)}
{(1, 1), (2, 2), (2, 1)}Input: n = 3
Output: 64
Explanation :
Reflexive Relation: A Relation R on A a set A is said to be Reflexive if xRx for every element of x ? A.
The number of reflexive relations on an n-element set is 2n(n-1)
How does this formula work?
A relation R is reflexive if the matrix diagonal elements are 1.
If we take a closer look the matrix, we can notice that the size of matrix is n2. The n diagonal entries are fixed. For remaining n2 – n entries, we have choice to either fill 0 or 1. So there are total 2n(n-1) ways of filling the matrix.
Below is the code implementation of the above approach:
// C++ Program to count reflexive relations // on a set of first n natural numbers. #include <iostream> using namespace std;
int countReflexive( int n)
{ // Return 2^(n*n - n)
return (1 << (n*n - n));
} int main()
{ int n = 3;
cout << countReflexive(n);
return 0;
} |
// Java Program to count reflexive // relations on a set of first n // natural numbers. import java.io.*;
import java.util.*;
class GFG {
static int countReflexive( int n)
{ // Return 2^(n*n - n) return ( 1 << (n*n - n));
} // Driver function public static void main (String[] args) {
int n = 3 ;
System.out.println(countReflexive(n));
}
} // This code is contributed by Gitanjali. |
# Python3 Program to count # reflexive relations # on a set of first n # natural numbers. def countReflexive(n):
# Return 2^(n*n - n)
return ( 1 << (n * n - n));
# driver function n = 3
ans = countReflexive(n);
print (ans)
# This code is contributed by saloni1297 |
// C# Program to count reflexive // relations on a set of first n // natural numbers. using System;
class GFG {
static int countReflexive( int n)
{
// Return 2^(n*n - n)
return (1 << (n*n - n));
}
// Driver function
public static void Main () {
int n = 3;
Console.WriteLine(countReflexive(n));
}
} // This code is contributed by vt_m. |
<?php // PHP Program to count // reflexive relations on a // set of first n natural numbers. function countReflexive( $n )
{ // Return 2^(n * n - n) return (1 << ( $n * $n - $n ));
} //Driver code $n = 3;
echo countReflexive( $n );
// This code is contributed by mits ?> |
<script> // Javascript Program to count reflexive
// relations on a set of first n
// natural numbers.
function countReflexive(n)
{
// Return 2^(n*n - n)
return (1 << (n*n - n));
}
let n = 3;
document.write(countReflexive(n));
// This code is contributed by divyesh072019.
</script> |
64
Time Complexity: O(1)
Auxiliary Space: O(1)