# Number of Irreflexive Relations on a Set

• Last Updated : 30 Apr, 2021

Given a positive integer N, the task is to find the number of irreflexive relations that can be formed over the given set of elements.  Since the count can be very large, print it to modulo 109 + 7.

A relation R on a set A is called reflexive if no (a, a) R holds for every element a € A.
For Example: If set A = {a, b} then R = {(a, b), (b, a)} is irreflexive relation.

Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

Examples:

Input: N = 2
Output: 4
Explanation:
Considering the set {1, 2}, the total possible irreflexive relations are:

• {}
• {(1, 2)}
• {(2, 1)}
• {(1, 2), (2, 1)}

Input: N = 5
Output: 1048576

Approach: Follow the steps below to solve the problem:

• A relation R on a set A is a subset of the cartesian product of a set, i.e. A * A with N2 elements.
• Irreflexive Relation: A relation R on a set A is called Irreflexive if and only if x R x [(x, x) does not belong to R] for every element x in A.
• There are total N pairs of (x, x) are present in the Cartesian product which should not be included in an irreflexive relation. Therefore, for the remaining (N2 – N) elements, each element has two choices i.e., either to include or exclude it in the subset.
• Hence, the total number of possible irreflexive relations is given by 2(N2 – N).

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include ``using` `namespace` `std;` `const` `int` `mod = 1000000007;` `// Function to calculate x^y``// modulo 1000000007 in O(log y)``int` `power(``long` `long` `x, unsigned ``int` `y)``{``    ``// Stores the result of x^y``    ``int` `res = 1;` `    ``// Update x if it exceeds mod``    ``x = x % mod;` `    ``// If x is divisible by mod``    ``if` `(x == 0)``        ``return` `0;` `    ``while` `(y > 0) {` `        ``// If y is odd, then``        ``// multiply x with result``        ``if` `(y & 1)``            ``res = (res * x) % mod;` `        ``// Divide y by 2``        ``y = y >> 1;` `        ``// Update the value of x``        ``x = (x * x) % mod;``    ``}` `    ``// Return the resultant value of x^y``    ``return` `res;``}` `// Function to count the number of``// irreflixive relations in a set``// consisting of N elements``int` `irreflexiveRelation(``int` `N)``{` `    ``// Return the resultant count``    ``return` `power(2, N * N - N);``}` `// Driver Code``int` `main()``{``    ``int` `N = 2;``    ``cout << irreflexiveRelation(N);` `    ``return` `0;``}`

## Java

 `// Java program for the above approach``import` `java.io.*;``import` `java.util.*;``class` `GFG``{` `static` `int` `mod = ``1000000007``;` `// Function to calculate x^y``// modulo 1000000007 in O(log y)``static` `int` `power(``int` `x, ``int` `y)``{``  ` `    ``// Stores the result of x^y``    ``int` `res = ``1``;` `    ``// Update x if it exceeds mod``    ``x = x % mod;` `    ``// If x is divisible by mod``    ``if` `(x == ``0``)``        ``return` `0``;` `    ``while` `(y > ``0``)``    ``{` `        ``// If y is odd, then``        ``// multiply x with result``        ``if` `((y & ``1``) != ``0``)``            ``res = (res * x) % mod;` `        ``// Divide y by 2``        ``y = y >> ``1``;` `        ``// Update the value of x``        ``x = (x * x) % mod;``    ``}` `    ``// Return the resultant value of x^y``    ``return` `res;``}` `// Function to count the number of``// irreflixive relations in a set``// consisting of N elements``static` `int` `irreflexiveRelation(``int` `N)``{` `    ``// Return the resultant count``    ``return` `power(``2``, N * N - N);``}`  `// Driver Code``public` `static` `void` `main(String[] args)``{``    ``int` `N = ``2``;``    ``System.out.println(irreflexiveRelation(N));``}``}` `// This code is contributed by code_hunt.`

## Python3

 `# Python3 program for the above approach``mod ``=` `1000000007` `# Function to calculate x^y``# modulo 1000000007 in O(log y)``def` `power(x, y):``    ``global` `mod``    ` `    ``# Stores the result of x^y``    ``res ``=` `1` `    ``# Update x if it exceeds mod``    ``x ``=` `x ``%` `mod` `    ``# If x is divisible by mod``    ``if` `(x ``=``=` `0``):``        ``return` `0` `    ``while` `(y > ``0``):` `        ``# If y is odd, then``        ``# multiply x with result``        ``if` `(y & ``1``):``            ``res ``=` `(res ``*` `x) ``%` `mod` `        ``# Divide y by 2``        ``y ``=` `y >> ``1` `        ``# Update the value of x``        ``x ``=` `(x ``*` `x) ``%` `mod` `    ``# Return the resultant value of x^y``    ``return` `res` `# Function to count the number of``# irreflixive relations in a set``# consisting of N elements``def` `irreflexiveRelation(N):` `    ``# Return the resultant count``    ``return` `power(``2``, N ``*` `N ``-` `N)` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``N ``=` `2``    ``print``(irreflexiveRelation(N))` `    ``# This code is contributed by mohit kumar 29.`

## C#

 `// C# program for above approach``using` `System;` `public` `class` `GFG``{` `  ``static` `int` `mod = 1000000007;` `  ``// Function to calculate x^y``  ``// modulo 1000000007 in O(log y)``  ``static` `int` `power(``int` `x, ``int` `y)``  ``{` `    ``// Stores the result of x^y``    ``int` `res = 1;` `    ``// Update x if it exceeds mod``    ``x = x % mod;` `    ``// If x is divisible by mod``    ``if` `(x == 0)``      ``return` `0;` `    ``while` `(y > 0)``    ``{` `      ``// If y is odd, then``      ``// multiply x with result``      ``if` `((y & 1) != 0)``        ``res = (res * x) % mod;` `      ``// Divide y by 2``      ``y = y >> 1;` `      ``// Update the value of x``      ``x = (x * x) % mod;``    ``}` `    ``// Return the resultant value of x^y``    ``return` `res;``  ``}` `  ``// Function to count the number of``  ``// irreflixive relations in a set``  ``// consisting of N elements``  ``static` `int` `irreflexiveRelation(``int` `N)``  ``{` `    ``// Return the resultant count``    ``return` `power(2, N * N - N);``  ``}` `  ``// Driver code``  ``public` `static` `void` `Main(String[] args)``  ``{``    ``int` `N = 2;``    ``Console.WriteLine(irreflexiveRelation(N));``  ``}``}` `// This code is contributed by sanjoy_62.`

## Javascript

 ``
Output:
`4`

Time Complexity: O(log N)
Auxiliary Space: O(1)

My Personal Notes arrow_drop_up