Number of Reflexive Relations on a Set

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 2^n(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 2^n(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)