The many faces of the XOR operation
XOR has always struck me as the odd one among the logical operators; since I first learned about it, I always felt a strange fascination with it. I think I was taught about it by my father, who cultivated my interest in computers and taught me programming when I was 7. I remember that the AND and OR operations were clear to me, and their use in the logic of a program; on the other hand, XOR seemed esoteric and not really useful, since I could easily program the same behavior with a combination of AND and OR. As I progressed in my programming journey and learned much more, I became aware of the strangeness and power of XOR. In this post, I explore the many interpretations of XOR, and why it deserves a unique appreciation.
The Boring Formal Definition™ (bear with me)
XOR, or Exclusive Or, is a binary logical operator that is True when only one of its two operands is True. It is usually represented with two symbols: ↮ and ⊕. I prefer ⊕. Its truth table is:
| A | B | A⊕B |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | F |
The basic interpretation: Or, with exclusion
In its most literal interpretation, XOR can be seen as an extension of OR. While OR is True when either of its operands is True -or when both are- XOR excludes this final case.
Inequality
XOR, when applied to boolean values, is equivalent to “not equal” ("!=" in C). XOR is True only when its two operands are different from each other. This interpretation reveals a profound truth about XOR: it is fundamentally about difference. XOR answers the question “are these values the same?” rather than “are these values true?”.
Toggle/Flip
XORing a boolean value with a constant True inverts the value/truthness of the value:
A ⊕ True = ¬A
If A is a binary number, this operation flips all the bits of A. We can use this to flip select bits in a variable, by setting True the corresponding bits in the second operand.
Programmable NOT
NOT is the unary logical operation that inverts the truth value of its operand. When the input is True, the output is False, and vice versa. XOR could be thought of as a version of NOT whose inverting behavior is controllable by a selector bit. When the second operand is True, the output is the negation of the first operand; When the second operand is False, the output remains equal to the first operand.
Addition
One of the most elegant uses of XOR is bridging between logic and arithmetic. If we take the truth table of XOR and substitute T with 1 and F with 0, we can see that the XOR operation is equivalent to arithmetic addition (without the carry). This makes it possible to create an electrical circuit that performs arithmetic addition on numbers represented in binary. This connection is not only theoretical: half-adders are circuits in real CPUs and use XOR gates to compute the sum bit, while the AND gate handles the carry.
Reversibility
XOR is its own self-inverse: (a ⊕ b) ⊕ b = a This unique property means XORing with the same value twice returns you to the original. This is profound because most operations aren’t reversible. This leads directly to some very interesting applications: encryption (one-time pad), in-place swapping without temporary variables, error correction codes.
Parity/Balance
XOR counts oddness: it returns True when an odd number of inputs are True. For multiple inputs: a ⊕ b ⊕ c ⊕ d tells you if there’s an odd number of True values. This interpretation is crucial for: parity bits in error detection, checksums, RAID systems.
Wrapping up
So there you have it: the many faces of XOR. I’m sure there are interpretations of XOR I haven’t discovered yet, patterns I haven’t recognized. That’s part of what keeps programming interesting: operations and concepts you thought you understood reveal new depths as you gain experience. What seemed like an esoteric curiosity when I was learning to program turned out to be a Swiss Army knife hiding in plain sight. It’s an inequality checker, a toggle switch, a half-adder, and a cryptographic building block all at once. My father gave me a gift when he introduced me to this strange operator: a gift made of curiosity and opportunity to be forever mesmerized by all the weird things that are out there, ready to be discovered.