XOR

Abstract

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$$B-A=\{x\in U:x\in B\wedge x\notin A\}$$

• Exclusive OR, not equal
• Returns true only when 2 inputs aren’t the same

Self-Inverse

• $a\oplus a=0$

XOR summation

Given a list of Integer (整数) $[a_1,a_2,\dots ,a_{n-1},a_n]$, let $x$ be the XOR summation of the list of integers $x=a_1\oplus a_2\dots \oplus a_{n-1}\oplus a_n$ , and then we put $x$ into the list. Shuffle the list randomly

Now when we pick the first element or a random element from the list, we are sure it is the XOR summation of the rest of the integers

Proof using Cases and Self-inverse, Own-Inverse and Commutativity (交换律). There are 2 possible outcomes of picking a random integer from the list

1. $x$, $x$ is the XOR summation by definition
2. $a_1$, $a_1$ is an integer that is in the given list

• For $a_1$ to be the XOR Summation of the rest of the elements, it must fulfil the following $a_1=a_2\oplus \dots \oplus a_{n-1}\oplus a_n\oplus x$
• We can expand the $x$ at the RHS, and we get $(a_2\oplus \dots \oplus a_{n-1}\oplus a_n)\oplus (a_1\oplus a_2\dots \oplus a_{n-1}\oplus a_n)$
• We can re-arrange the RHS with commutativity, and we get $a_1\oplus a_2\oplus a_2\dots \oplus a_{n-1}\oplus a_{n-1}\oplus a_n\oplus a_n$
• With Self-Inverse and Own-Inverse, we can reduce the RHS to $a_1$
• Since RHS is equal to LHS, $a_1=a_2\dots \oplus a_{n-1}\oplus a_n\oplus x$ is valid. Thus $a_1$ is the XOR summation of the rest of the elements

Practice problem: Codeforces - XOR Mixup

Own-Inverse

• $a\oplus 0=a$

Chemistry with Self-Inverse

Given $a\oplus b$, we can get back $a$ by $a\oplus b\oplus b$ . Because $b\oplus b=0$ based on Self-Inverse, and $a\oplus 0=a$ based on Own-Inverse

Swapping values without introducing a new variable

public class MyClass {
    public static void main(String args[]) {
      int x=10;
      int y=25;
      
      x = x ^ y; // x ^ x = y, x ^ y = x
      y = x ^ y; // x
      x = x ^ y; // y
 
      System.out.println("x: " + x);
      System.out.println("y: " + y);
    }
}

Logic Gate Implementation

The Mathematical Statement is $(A\vee B)\wedge (\neg A\vee \neg B)$

1. OR: $A$ or $B$ must be true
2. AND, NOT (De Morgan’s laws): Either $A$ or $B$ are false
3. AND: The above 2 must be true in order to let the statement to be true