Given a set, find XOR of the XOR’s of all subsets.

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

<?php
// PHP program to find XOR
// of XOR's of all subsets

// Returns XOR of all
// XOR's of given subset
function 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 = array(1, 2, 3);
    $n = count($Set);
    echo "XOR of XOR's of all subsets is "
         , findXOR($Set, $n);

// This code is contributed by anuj_67.
?>

Javascript

<script>

// JavaScript program to find XOR of XOR's of all subsets

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

// Driver program

    let Set = [1, 2, 3];
    let n = Set.length;
    document.write("XOR of XOR's of all subsets is "
        + findXOR(Set, n));

// This code is contributed by Surbhi Tyagi

</script>

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.
This article is contributed by Ekta Goel. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.