Given a set, find XOR of the XOR's of all subsets. (original) (raw)

Last Updated : 23 Jul, 2025

The question is to find XOR of the XOR's of all subsets. i.e if the set is {1,2,3}. All subsets are : [{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}]. Find the XOR of each of the subset and then find the XOR of every subset result.
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This is a very simple question to solve if we get the first step (and only step) right. The solution is XOR is always 0 when n > 1 and Set[0] when n is 1.

C++ `

// C++ program to find XOR of XOR's of all subsets #include <bits/stdc++.h> using namespace std;

// Returns XOR of all XOR's of given subset int findXOR(int Set[], int n) { // XOR is 1 only when n is 1, else 0 if (n == 1) return Set[0]; else return 0; }

// Driver program int main() { int Set[] = { 1, 2, 3 }; int n = sizeof(Set) / sizeof(Set[0]); cout << "XOR of XOR's of all subsets is " << findXOR(Set, n); return 0; }

C

// C program to find the XOR of XORs of all subsets #include <stdio.h>

// Returns XOR of all XORs of given subset int findXOR(int Set[], int n) { // XOR is 1 only when n is 1, else 0 if (n == 1) return Set[0]; else return 0; }

// Driver program int main() { int Set[] = { 1, 2, 3 }; int n = sizeof(Set) / sizeof(Set[0]); printf("XOR of XORs of all subsets is %d\n", findXOR(Set, n)); return 0; }

// This code is contributed by phalashi.

Java

// Java program to find XOR of // XOR's of all subsets import java.util.*;

class GFG {

// Returns XOR of all XOR's of given subset
static int findXOR(int Set[], int n)
{

    // XOR is 1 only when n is 1, else 0
    if (n == 1)
        return Set[0];
    else
        return 0;
}

// Driver code
public static void main(String arg[])
{
    int Set[] = { 1, 2, 3 };
    int n = Set.length;
    System.out.print("XOR of XOR's of all subsets is "
                     + findXOR(Set, n));
}

}

// This code is contributed by Anant Agarwal.

Python3

Python program to find

XOR of XOR's of all subsets

Returns XOR of all

XOR's of given subset

def findXOR(Set, n):

# XOR is 1 only when
# n is 1, else 0
if (n == 1):
    return Set[0]
else:
    return 0

Driver code

Set = [1, 2, 3] n = len(Set)

print("XOR of XOR's of all subsets is ", findXOR(Set, n))

This code is contributed

by Anant Agarwal.

C#

// C# program to find XOR of // XOR's of all subsets using System;

class GFG {

// Returns XOR of all
// XOR's of given subset
static int findXOR(int[] Set, int n)
{

    // XOR is 1 only when n
    // is 1, else 0
    if (n == 1)
        return Set[0];
    else
        return 0;
}

// Driver code
public static void Main()
{
    int[] Set = { 1, 2, 3 };
    int n = Set.Length;
    Console.Write("XOR of XOR's of all subsets is "
                  + findXOR(Set, n));
}

}

// This code is contributed by nitin mittal

PHP

JavaScript

`

Output:

XOR of XOR's of all subsets is 0

Time Complexity: O(1)

Auxiliary Space: O(1)

Related Problem :
Sum of XOR of all possible subsets
How does this work?
The logic goes simple. When we apply XOR on all the subsets of a set, we can use the commutative and associative property of XOR which reduces the problem to finding XOR result of each element that depends on the total number of occurrences of each element. Eg. XOR([{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}]) = XOR(XOR(1,1,1,1), XOR(2,2,2,2), XOR(3,3,3,3), XOR(4,4,4,4))

Let us consider n'th element, it can be included in the power set of remaining (n-1) elements. The number of subsets for (n-1) elements is equal to 2(n-1) which is always even when n>1. Thus, in the XOR result, every element is included even number of times and XOR of even occurrences of any number is 0.