In this article, we will explain ways to find reflexive relationships on a set. In this problem, we are given a number n, and a set of n natural numbers, and we must determine the number of reflexive relations.
Reflexive relation - If for every 'a' in set A, (a, a) belongs to relation R, then relation R is said to be a reflexive relation on set A. For example -
Input : x = 1 Output : 1 Explanation : set = { 1 }, reflexive relations on A * A : { { 1 } } Input : x = 2 Output : 4 Explanation : set = { 1,2 }, 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 ) }
Thus, if for every element a ∈ A, there is (a, a) ∈ R, then the relation R is reflexive.
Solution method
The number of reflexive relations on the element set can be calculated through the formula 2n2−n. This general formula is obtained by counting the number of reflexive relations of integers.
Example
#include <iostream> using namespace std; int countReflexive(int n){ int ans = 1 << (n*n - n); return ans; } int main(){ int n ; cin >> n ; // taking input n from the user using std cin. int result = countReflexive(n); // calling function to calculate number of reflexive relations cout << "Number of reflexive relations on set: " << result ; // printing the answer return 0; }
Output
Number of reflexive relations on set: 1
Explanation of the above program
This program is easy to understand because we just Get the input from the user and put it into the formula 2n2−n. We use the left shift operator "
Conclusion
In this paper, we addressed a problem about the number of reflexive relations on sets. We discussed simple ways to solve a given problem, and mathematicians derived a formula for counting the number of reflexive relations.
We also learned to write a program for this problem in C, with a time complexity of O(1). We can write the same program in other languages like C, Java, Python and others.
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