The logical operation exclusive disjunction, also called exclusive or (symbolized XOR, EOR, or ⊕), is a type of logical disjunction on two operands that results in a value of true if exactly one of the operands has a value of true.^{[1]} A simple way to state this is "one or the other but not both."
Put differently, exclusive disjunction is a logical operation on two logical values, typically the values of two propositions, that produces a value of true only in cases where the truth value of the operands differ.
If both operands are the same, the effect is to set all bits to 0. This is a common method of setting a memory location to hexadecimal zeroes.
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The truth table of (also written as , or ) is as follows:
p  q  ⊕ 

F  F  F 
F  T  T 
T  F  T 
T  T  F 
Note the threeway symmetry of the outcomes: The column headings p, q, and in this table could be arbitrarily reassigned, and the table would still be correct.
The Venn diagram of (red part is true)
The following equivalents can then be deduced, written with logical operators, in mathematical and engineering notation:
Generalized or nary XOR is true when the number of 1bits is odd.
The exclusive disjunction can be expressed in terms of the logical conjunction (), the disjunction (), and the negation () as follows:
The exclusive disjunction can also be expressed in the following way:
This representation of XOR may be found useful when constructing a circuit or network, because it has only one operation and small number of and operations. The proof of this identity is given below:
It is sometimes useful to write in the following way:
This equivalence can be established by applying De Morgan's laws twice to the fourth line of the above proof.
The exclusive or is also equivalent to the negation of a logical biconditional, by the rules of material implication (a material conditional is equivalent to the disjunction of the negation of its antecedent and its consequence) and material equivalence.
Although the operators (conjunction) and (disjunction) are very useful in logic systems, they fail a more generalizable structure in the following way:
The systems and are monoids. This unfortunately prevents the combination of these two systems into larger structures, such as a mathematical ring.
However, the system using exclusive or is an abelian group. The combination of operators and over elements {T,F} produce the wellknown field F_{2}. This field can represent any logic obtainable with the system and has the added benefit of the arsenal of algebraic analysis tools for fields.
The Oxford English Dictionary explains “either … or” as follows:
The exclusiveor explicitly states “one or the other but not neither nor both.”
Following this kind of commonsense intuition about “or”, it is sometimes argued that in many natural languages, English included, the word “or” has an “exclusive” sense. The exclusive disjunction of a pair of propositions, (p, q), is supposed to mean that p is true or q is true, but not both. For example, it is argued, the normal intention of a statement like “You may have coffee or you may have tea” is to stipulate that exactly one of the conditions can be true. Certainly under many circumstances a sentence like this example should be taken as forbidding the possibility of one's accepting both options. Even so, there is good reason to suppose that this sort of sentence is not disjunctive at all. If all we know about some disjunction is that it is true overall, we cannot be sure that either of its disjuncts is true. For example, if a woman has been told that her friend is either at the snack bar or on the tennis court, she cannot validly infer that he is on the tennis court. But if her waiter tells her that she may have coffee or she may have tea, she can validly infer that she may have tea. Nothing classically thought of as a disjunction has this property. This is so even given that she might reasonably take her waiter as having denied her the possibility of having both coffee and tea.
(Note: If the waiter intends that choosing neither tea nor coffee is an option i.e. ordering nothing, the appropriate operator is NAND: p NAND q.)
There are also good general reasons to suppose that no word in any natural language could be adequately represented by the binary exclusive “or” of formal logic. First, any binary or other nary exclusive “or” is true if and only if it has an odd number of true inputs. But it seems as though no word in any natural language that can conjoin a list of two or more options has this general property. Second, as pointed out by Barrett and Stenner in the 1971 article “The Myth of the Exclusive ‘Or’” (Mind, 80 (317), 116–121), no author has produced an example of an English orsentence that appears to be false because both of its inputs are true. Certainly there are many orsentences such as “The light bulb is either on or off” in which it is obvious that both disjuncts cannot be true. But it is not obvious that this is due to the nature of the word “or” rather than to particular facts about the world.
The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphasized in a given context of discussion. In addition to the abbreviation “XOR”, any of the following symbols may also be seen:
p  q  p + q 

0  0  0 
0  1  1 
1  0  1 
1  1  0 
^
) is used to denote the bitwise XOR operator. This is not used outside of programming contexts because it is too easily confused with other uses of the caret.This section uses the following symbols:
The following equations follow from logical axioms:
In view of the isomorphism between addition modulo 2 and exclusive disjunction, it is clear that XOR is both an associative and a commutative operation. Thus parentheses may be omitted in successive operations and the order of terms makes no difference to the result. For example, we have the following equations:
Exclusive disjunction is often used for bitwise operations. Examples:
As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two nbit strings is identical to the standard vector of addition in the vector space .
In computer science, exclusive disjunction has several uses:
In logical circuits, a simple adder can be made with an XOR gate to add the numbers, and a series of AND, OR and NOT gates to create the carry output.
On some computer architectures, it is more efficient to store a zero in a register by xoring the register with itself (bits xored with themselves are always zero) instead of loading and storing the value zero.
In simple threshold activated neural networks, modeling the ‘xor’ function requires a second layer because ‘xor’ is not a linearlyseparable function.
Exclusiveor is sometimes used as a simple mixing function in cryptography, for example, with onetime pad or Feistel network systems.
XOR is used in RAID 3–6 for creating parity information. For example, RAID can “back up” bytes 10011100
and 01101100
from two (or more) hard drives by XORing (11110000
) and writing to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be recreated by XORing bytes from the remaining drives. If the drive containing 01101100
is lost, 10011100
and 11110000
can be XORed to recover the lost byte.
XOR is also used to detect an overflow in the result of a signed binary arithmetic operation. If the leftmost retained bit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORing those two bits will give a “1” if there is an overflow.
XOR can be used to swap two numeric variables in computers, using the XOR swap algorithm; however this is regarded as more of a curiosity and not encouraged in practice.
In computer graphics, XORbased drawing methods are often used to manage such items as bounding boxes and cursors on systems without alpha channels or overlay planes.

Contents 
XOR
Infinitive 
Third person singular 
Simple past 
Past participle 
Present participle 
to XOR (thirdperson singular simple present XORs, present participle XORing, simple past and past participle XORed)

