XOR gate

This article is about XOR in the sense of an electronic logic gate (e.g. CMOS 4030). For XOR in the purely logical sense, see Exclusive disjunction. For other uses, see XOR (disambiguation).

INPUT A   B		OUTPUT A XOR B
0	0	0
0	1	1
1	0	1
1	1	0

The XOR gate (sometimes EOR gate, or EXOR gate) is a digital logic gate that implements an exclusive or; that is, a true output (1) results if one, and only one, of the inputs to the gate is true (1). If both inputs are false (0) or both are true (1), a false output (0) results. Its behavior is summarized in the truth table shown on the right. A way to remember XOR is "one or the other but not both". It represents the inequality function, i.e., the output is HIGH (1) if the inputs are not alike otherwise the output is LOW (0).

This function is addition modulo 2. As a result, XOR gates are used to implement binary addition in computers. A half adder consists of an XOR gate and an AND gate.

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There are two symbols for XOR gates: the 'military' symbol which was originally developed in America and the 'rectangular' symbol, which was originally British. For more information see Logic Gate Symbols.

'ANSI' "Military" XOR Symbol

'IEC' "Rectangular" XOR Symbol

The XOR gate with inputs A and B implements the logical expression $A\cdot {\overline {B}}+{\overline {A}}\cdot B$.

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If a specific type of gate is not available, it can be constructed from other available gates. An XOR gate can be trivially constructed from an XNOR gate followed by a NOT gate. If we consider the expression $A\cdot {\overline {B}}+{\overline {A}}\cdot B$, we can construct an XOR gate directly using AND, OR and NOT gates. However, this approach requires four gates of three different kinds.

An XOR gate can be made from four NAND or five NOR gates in the configurations shown below. In fact, both NAND and NOR gates are so-called "universal gates," and any logical function can be constructed from either NAND logic or NOR logic alone.

XOR gate constructed using only NAND gates.

More than two inputs

Strict reading of the definition of exclusive or, or observation of the IEC symbol, raises the question of correct behaviour with additional inputs. If a logic gate were to accept three or more inputs and produce a true output if exactly one of those inputs were true, then it would in effect be a one-hot detector (and indeed this is the case for only two inputs). However, it is rarely implemented this way in practice.

It is most common to regard subsequent inputs as being applied through a cascade of binary exclusive-or operations: the first two signals are fed into an XOR gate, then the output of that gate is fed into a second XOR gate together with the third signal, and so on for any remaining signals. The result is a circuit that outputs a 1 when the number of 1s at its inputs is odd, and a 0 when the number of incoming 1s is even. This makes it practically useful as a parity generator or a modulo-2 adder.

For example, the 74LVC1G386 microchip is advertised as a three-input logic gate, and implements a parity generator.

Applications

Uses in Addition

The XOR logic gate can be used as a one-bit adder that adds any two bits together to output one bit. For example, if we add 1 plus 1 in binary, we expect a two-bit answer, 10 (i.e. 2 in decimal). Since the trailing sum bit in this output is achieved with XOR, the preceding carry bit is calculated with an AND gate. This is the main principle in Half Adders and the combined AND-XOR circuit may be chained together in order to add ever longer binary numbers.

Example half adder circuit diagram

Signal crossing

XOR gates can be used to cross two Boolean signals without requiring a via. The crossover comprises three XOR gates, strategically positioned to cancel out the effects of each other. A similar result can be achieved using eight NAND gates. Note that this setup can only cross Boolean (digital) signals, and not analog signals, so would be of limited use in an amplifier or other analog circuit.

Crossover comprising three XOR gates

Pseudo-random number generation

Pseudo-random number generators, specifically Linear feedback shift registers, are defined in terms of the exclusive-or operation. Hence, a suitable setup of XOR gates can model a linear feedback shift register, in order to generate random sequences of bits.

See also

• Exclusive or
• XNOR gate
• Boolean algebra (logic)
• Logic gates