Exclusive disjunction, also known as exclusive or and symbolized by XOR or EOR, is a logical operation on two operands that results in a logical value of true if and only if one of the operands, but not both, has a value of true. In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...

XOR Logic Gate

XOR Logic Gate Symbol Remark: Accidentially uploaded the symbol for XNOR here first. ... XOR Logic Gate Symbol Remark: Accidentially uploaded the symbol for XNOR here first. ... A logic gate is an arrangement of controlled switches used to calculate operations using Boolean logic in digital circuits. ...

Definition

In many natural languages, English included, the interpretation of the word "or" requires a certain amount of care. The exclusive disjunction of a pair of propositions, (p, q), means that p is true or q is true, but not both. For example, the normal intention of a statement like "You can follow the rules or be disqualified" is to stipulate that exactly one of the conditions can be true. In logic, however, the default meaning of the word "or" is the inclusive disjunction, which signifies that at least one of the alternatives is true. Other languages, such as Latin, may have different words for these different types of "or". The English language is a West Germanic language that originates in England. ... OR logic gate. ... Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ...

Exclusive disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if one but not both of its operands is true. In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... Proposition is a term used in logic to describe the content of assertions. ...

The truth table of p XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

The following equivalents can then be deduced:

Alternative symbols

The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphaisized in a given context of discussion. In addition to the abbreviation "XOR", any of the following symbols may also be seen:

• A plus sign (+). This makes sense mathematically because exclusive disjunction corresponds to addition modulo 2, which has the following addition table, clearly isomorphic to the one above:

• The use of the plus sign has the added advantage that all of the ordinary algebraic properties of mathematical rings and fields can be used without further ado.
• A plus sign that is modified in some way, such as being encircled (⊕}. This usage faces the objection that this same symbol is already used in mathematics for the direct sum of algebraic structures.
• An inclusive disjunction symbol (∨) that is modified in some way, such as being underlined (∨).
• In the C programming language and the Java programming language, a caret (^) is used to denote both the ordinary XOR and the bitwise XOR operator. This is not used outside of programming contexts because it is too easily confused with other uses of the caret.
• The symbol .

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Equivalent expressions

The exclusive disjunction can be expressed in terms of the 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 p XOR q in the following way:

This equivalence can be established by applying De Morgan's laws twice to the fourth line of the above proof. note that demorgans laws are also a big part in circut design. ...

Associativity and commutativity

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: In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

Properties

This section uses the following symbols:

The following equations follow from logical axioms:

Bitwise operation

Exclusive disjunction is often used for bitwise operations. Examples:

• 1 xor 1 = 0
• 1 xor 0 = 1
• 1110 xor 1001 = 0111 (this is equivalent to addition without carry)

The term carry may refer to: A violation whilst dribbling in the game of basketball. ...

Computer science

In computer science, exclusive disjunction has several uses:

• It tells whether two bits are unequal.
• It is an optional bit-flipper (the deciding input chooses whether to invert the data input).
• It tells whether there are an odd number of 1 bits (A ⊕ B ⊕ C ⊕ D ⊕ E is true iff an odd number of the variables are true).

On some computer architectures, it is more efficient to store a zero in a register by xor-ing the register with itself (bits xor-ed with themselves are always zero) instead of loading and storing the value zero. Because 'xor' is a more complex logical function than 'or' and 'and', its neural network requires an additional processing layer. Negation (i. ... In mathematics, the parity of an object refers to whether is is even or odd. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In computer science, efficiency is used to describe several desirable properties of an algorithm or other construct, besides clean design, functionality, etc. ... Simplified view of an artificial neural network A neural network is an interconnected group of biological neurons. ...

Exclusive-or is sometimes used as a simple mixing function in cryptography, for example, with one-time pad or Feistel network systems. The German Lorenz cipher machine Cryptography or cryptology is a field of mathematics and computer science concerned with information security and related issues, particularly encryption and authentication. ... Excerpt from a one-time pad. ... In cryptography, a Feistel cipher is a block cipher with a particular structure, named after IBM cryptographer Horst Feistel; it is also commonly known as a Feistel network. ...

XOR is used in RAID 3–6 for creating parity information. For example, RAID can "back up" bytes 10011100 and 01101100 from two hard drives by XORing (11110000) and writing to a third drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-created 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. In computing, a redundant array of inexpensive disks, also later known as redundant array of independent disks (commonly abbreviated RAID) is a system which uses multiple hard drives to share or replicate data among the drives. ...

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 "one" if there is an overflow.

See also

Other operators

AND Logic Gate In mathematics, logical conjunction (usual symbol and) is a logical operator that results in false if either of the operands is false. ... OR logic gate. ... XNOR Logic Gate Symbol Logical equality is a logical operator that corresponds to equality in boolean algebra and to the logical biconditional in propositional calculus. ... In propositional calculus, or logical calculus in mathematics, the material conditional or the implies operator is a binary truth-functional logical operator yielding the form If a then c, where a and c are statement variables (to be replaced by any meaningful indicative sentence of the language). ... NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ... NOR Logic Gate Logical nor (not or), joint denial, or Webb-operation is a boolean logic operator which produces a result that is the inverse of logical or. ...

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