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53-618: Exor may refer to: Exclusive or , the exclusive disjunction Exor , antagonist in Super Mario RPG ExOR (wireless network protocol) , a protocol for a wireless ad-hoc networks Exor (company) , an Italian investment holding company based in the Netherlands controlled by the Agnelli family See also [ edit ] XOR (disambiguation) Topics referred to by

106-468: A bit in a binary number , truth table values can be efficiently encoded as integer values in electronic design automation (EDA) software . For example, a 32-bit integer can encode the truth table for a LUT with up to 5 inputs. When using an integer representation of a truth table, the output value of the LUT can be obtained by calculating a bit index k based on the input values of the LUT, in which case

159-405: A combination of input values for the variables A and B. These combinations now can be combined with the output of the function corresponding to that combination, thus forming the set of input-output pairs as a special relation that is a subset of A×F. For a relation to be a function, the special requirement is that each element of the domain of the function must be mapped to one and only one member of

212-472: A random bit XORed with a non-random bit will result in a random bit. Multiple sources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteed to be at least as good as the best individual source. XOR is used in RAID 3–6 for creating parity information. For example, RAID can "back up" bytes 10011100 2 and 01101100 2 from two (or more) hard drives by XORing

265-486: 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 XOR-ing the register with itself (bits XOR-ed with themselves are always zero) than to load and store the value zero. In cryptography , XOR is sometimes used as a simple, self-inverse mixing function, such as in one-time pad or Feistel network systems. XOR

318-528: A simple and straightforward way to encode Boolean functions, however given the exponential growth in size as the number of inputs increase, they are not suitable for functions with a large number of inputs. Other representations which are more memory efficient are text equations and binary decision diagrams . In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic Boolean operations to simple correlations of inputs to outputs, without

371-457: A tabular format, in which each row corresponds to a member of the domain paired with its corresponding output value, 0 or 1. Of course, for the Boolean functions, we do not have to list all the members of the domain with their images in the codomain ; we can simply list the mappings that map the member to "1", because all the others will have to be mapped to "0" automatically (that leads us to

424-601: A truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883–84 in connection with the composition of Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation" that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for

477-411: Is 2×2, or four. So the result is four possible outputs of C and R. If one were to use base 3, the size would increase to 3×3, or nine possible outputs. The first "addition" example above is called a half-adder. A full-adder is when the carry from the previous operation is provided as input to the next adder. Thus, a truth table of eight rows would be needed to describe a full adder 's logic: Regarding

530-557: Is a group . This unfortunately prevents the combination of these two systems into larger structures, such as a mathematical ring . However, the system using exclusive or ( { T , F } , ⊕ ) {\displaystyle (\{T,F\},\oplus )} is an abelian group . The combination of operators ∧ {\displaystyle \wedge } and ⊕ {\displaystyle \oplus } over elements { T , F } {\displaystyle \{T,F\}} produce

583-403: Is a mathematical table used in logic —specifically in connection with Boolean algebra , Boolean functions , and propositional calculus —which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables . In particular, truth tables can be used to show whether a propositional expression

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636-530: Is also called "not left-right arrow" ( \nleftrightarrow ) in LaTeX -based markdown ( ↮ {\displaystyle \nleftrightarrow } ). Apart from the ASCII codes, the operator is encoded at U+22BB ⊻ XOR ( ⊻ ) and U+2295 ⊕ CIRCLED PLUS ( ⊕, ⊕ ), both in block mathematical operators . Truth table A truth table

689-557: Is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as French soit... soit . 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: If using binary values for true (1) and false (0), then exclusive or works exactly like addition modulo 2. Exclusive disjunction

742-497: Is also heavily used in block ciphers such as AES (Rijndael) or Serpent and in block cipher implementation (CBC, CFB, OFB or CTR). In simple threshold-activated artificial neural networks , modeling the XOR function requires a second layer because XOR is not a linearly separable function. Similarly, XOR can be used in generating entropy pools for hardware random number generators . The XOR operation preserves randomness, meaning that

795-429: Is an operation on one logical value p, for which the output value remains p. The truth table for the logical identity operator is as follows: Logical negation is an operation on one logical value , typically the value of a proposition , that produces a value of true if its operand is false and a value of false if its operand is true. The truth table for NOT p (also written as ¬p , Np , Fpq , or ~p )

848-478: Is called the function's algebraic normal form . Disjunction is often understood exclusively in natural languages . In English, the disjunctive word "or" is often understood exclusively, particularly when used with the particle "either". The English example below would normally be understood in conversation as implying that Mary is not both a singer and a poet. However, disjunction can also be understood inclusively, even in combination with "either". For instance,

901-410: Is different from Wikidata All article disambiguation pages All disambiguation pages Exclusive or It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true. XOR excludes that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B". It is symbolized by

954-550: Is equivalent to the disjunction of the negation of its antecedent and its consequence) and material equivalence . In summary, we have, in mathematical and in engineering notation: By applying the spirit of De Morgan's laws , we get: ¬ ( p ↮ q ) ⇔ ¬ p ↮ q ⇔ p ↮ ¬ q . {\displaystyle \lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.} Although

1007-722: Is false, the output is X {\displaystyle X} , and when G {\displaystyle G} is true, the output is ¬ X {\displaystyle \lnot X} . The function table for this would look like: Similarly, a 4-to-1 multiplexer with select imputs S 0 {\displaystyle S_{0}} and S 1 {\displaystyle S_{1}} , data inputs A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} and D {\displaystyle D} , and output Z {\displaystyle Z} (as displayed in

1060-469: 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 n -bit strings is identical to the standard vector of addition in the vector space ( Z / 2 Z ) n {\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n}} . In computer science, exclusive disjunction has several uses: In logical circuits,

1113-398: Is sometimes useful to write p ↮ q {\displaystyle p\nleftrightarrow q} in the following way: or: 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

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1166-409: Is true for all legitimate input values, that is, logically valid . A truth table has one column for each input variable (for example, A and B), and one final column showing all of the possible results of the logical operation that the table represents (for example, A XOR B ). Each row of the truth table contains one possible configuration of the input variables (for instance, A=true, B=false), and

1219-428: Is useful especially if the operations are commutative, although one can additionally specify that the rows are the first operand and the columns are the second operand. This condensed notation is particularly useful in discussing multi-valued extensions of logic, as it significantly cuts down on combinatoric explosion of the number of rows otherwise needed. It also provides for quickly recognizable characteristic "shape" of

1272-548: The guide columns to the left of a table, which represent propositional variables , different authors have different recommendations about how to fill them in, although this is of no logical significance. Lee Archie, a professor at Lander University , recommends this procedure, which is commonly followed in published truth-tables: This method results in truth-tables such as the following table for " P ⊃ (Q ∨ R ⊃ (R ⊃ ¬P)) ", produced by Stephen Cole Kleene : Colin Howson , on

1325-473: The k th bit of the binary representation of the truth table is the LUT's output value, where k = V 0 × 2 0 + V 1 × 2 1 + V 2 × 2 2 + ⋯ + V n × 2 n {\displaystyle k=V_{0}\times 2^{0}+V_{1}\times 2^{1}+V_{2}\times 2^{2}+\dots +V_{n}\times 2^{n}} . Truth tables are

1378-423: The logical conjunction ("logical and", ∧ {\displaystyle \wedge } ), the disjunction ("logical or", ∨ {\displaystyle \lor } ), and the negation ( ¬ {\displaystyle \lnot } ) as follows: The exclusive disjunction p ↮ q {\displaystyle p\nleftrightarrow q} can also be expressed in

1431-427: The minterms idea). Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table in his Tractatus Logico-Philosophicus , which was completed in 1918 and published in 1921. Such a system was also independently proposed in 1921 by Emil Leon Post . Irving Anellis 's research shows that C.S. Peirce appears to be the earliest logician (in 1883) to devise a truth table matrix. From

1484-529: The operators ∧ {\displaystyle \wedge } ( conjunction ) and ∨ {\displaystyle \lor } ( disjunction ) are very useful in logic systems, they fail a more generalizable structure in the following way: The systems ( { T , F } , ∧ ) {\displaystyle (\{T,F\},\wedge )} and ( { T , F } , ∨ ) {\displaystyle (\{T,F\},\lor )} are monoids , but neither

1537-444: The 7 most commonly used out of the 16 possible truth functions of two Boolean variables P and Q: where  T  means true and  F  means false For binary operators, a condensed form of truth table is also used, where the row headings and the column headings specify the operands and the table cells specify the result. For example, Boolean logic uses this condensed truth table notation: This notation

1590-426: The LUT's output value is the k th bit of the integer. For example, to evaluate the output value of a LUT given an array of n Boolean input values, the bit index of the truth table's output value can be computed as follows: if the i th input is true, let V i = 1 {\displaystyle V_{i}=1} , else let V i = 0 {\displaystyle V_{i}=0} . Then

1643-536: The above have motivated analyses of the exclusivity inference as pragmatic conversational implicatures calculated on the basis of an inclusive semantics . Implicatures are typically cancellable and do not arise in downward entailing contexts if their calculation depends on the Maxim of Quantity . However, some researchers have treated exclusivity as a bona fide semantic entailment and proposed nonclassical logics which would validate it. This behavior of English "or"

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1696-432: The codomain. Thus, the function f itself can be listed as: f = {((0, 0), f 0 ), ((0, 1), f 1 ), ((1, 0), f 2 ), ((1, 1), f 3 )}, where f 0 , f 1 , f 2 , and f 3 are each Boolean, 0 or 1, values as members of the codomain {0, 1}, as the outputs corresponding to the member of the domain, respectively. Rather than a list (set) given above, the truth table then presents these input-output pairs in

1749-417: The conditional. Truth tables can be used to prove many other logical equivalences . For example, consider the following truth table: This demonstrates the fact that p ⇒ q {\displaystyle p\Rightarrow q} is logically equivalent to ¬ p ∨ q {\displaystyle \lnot p\lor q} . Here is a truth table that gives definitions of

1802-416: The distribution of the values in the table which can assist the reader in grasping the rules more quickly. Truth tables are also used to specify the function of hardware look-up tables (LUTs) in digital logic circuitry . For an n-input LUT, the truth table will have 2^ n values (or rows in the above tabular format), completely specifying a Boolean function for the LUT. By representing each Boolean value as

1855-406: The first example below shows that "either" can be felicitously used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away under downward entailing contexts. If disjunction were understood as exclusive in this example, it would leave open the possibility that some people ate both rice and beans. Examples such as

1908-399: The following way: This representation of XOR may be found useful when constructing a circuit or network, because it has only one ¬ {\displaystyle \lnot } operation and small number of ∧ {\displaystyle \land } and ∨ {\displaystyle \lor } operations. A proof of this identity is given below: It

1961-557: The image) would have this function table: Here is an extended truth table giving definitions of all sixteen possible truth functions of two Boolean variables p and q : where In proposition 5.101 of the Tractatus Logico-Philosophicus , Wittgenstein listed the table above as follows: The truth table represented by each row is obtained by appending the sequence given in Truthvalues row to

2014-596: The inputs differ: Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true if and only if one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction p ↮ q {\displaystyle p\nleftrightarrow q} , also denoted by p ? ⁡ q {\displaystyle p\operatorname {?} q} or J p q {\displaystyle Jpq} , can be expressed in terms of

2067-406: The just mentioned bytes, resulting in ( 11110000 2 ) and writing it to another 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. For instance, if the drive containing 01101100 2 is lost, 10011100 2 and 11110000 2 can be XORed to recover the lost byte. XOR is also used to detect an overflow in

2120-404: The logical "AND" operation as multiplication on F 2 {\displaystyle \mathbb {F} _{2}} and the "XOR" operation as addition on F 2 {\displaystyle \mathbb {F} _{2}} : The description of a Boolean function as a polynomial in F 2 {\displaystyle \mathbb {F} _{2}} , using this basis,

2173-653: The number of different functions of n variables is the double exponential 2 . Truth tables for functions of three or more variables are rarely given. It can be useful to have the output of a truth table expressed as a function of some variable values, instead of just a literal truth or false value. These may be called "function tables" to differentiate them from the more general "truth tables". For example, one value, G {\displaystyle G} , may be used with an XOR gate to conditionally invert another value, X {\displaystyle X} . In other words, when G {\displaystyle G}

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2226-483: The other hand, believes that "it is a good practical rule" to do the following: to start with all Ts, then all the ways (three) two Ts can be combined with one F, then all the ways (three) one T can be combined with two Fs, and then finish with all Fs. If a compound is built up from n distinct sentence letters, its truth table will have 2 rows, since there are two ways of assigning T or F to the first letter, and for each of these there will be two ways of assigning T or F to

2279-417: The output belongs to a binary set, i.e. F = {0, 1}. For an n-ary Boolean function, the inputs come from a domain that is itself a Cartesian product of binary sets corresponding to the input Boolean variables. For example for a binary function, f (A, B), the domain of f is A×B, which can be listed as: A×B = {(A = 0, B = 0), (A = 0, B = 1), (A = 1, B = 0), (A = 1, B = 1)}. Each element in the domain represents

2332-854: The prefix operator J {\displaystyle J} and by the infix operators XOR ( / ˌ ɛ k s ˈ ɔː r / , / ˌ ɛ k s ˈ ɔː / , / ˈ k s ɔː r / or / ˈ k s ɔː / ), EOR , EXOR , ∨ ˙ {\displaystyle {\dot {\vee }}} , ∨ ¯ {\displaystyle {\overline {\vee }}} , ∨ _ {\displaystyle {\underline {\vee }}} , ⩛ , ⊕ {\displaystyle \oplus } , ↮ {\displaystyle \nleftrightarrow } , and ≢ {\displaystyle \not \equiv } . The truth table of A ⊕ B {\displaystyle A\oplus B} shows that it outputs true whenever

2385-777: 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. XOR linked lists leverage XOR properties in order to save space to represent doubly linked list data structures. In computer graphics , XOR-based drawing methods are often used to manage such items as bounding boxes and cursors on systems without alpha channels or overlay planes. It

2438-415: The result of the operation for those values. A truth table is a structured representation that presents all possible combinations of truth values for the input variables of a Boolean function and their corresponding output values. A function f from A to F is a special relation , a subset of A×F, which simply means that f can be listed as a list of input-output pairs. Clearly, for the Boolean functions,

2491-560: The result, this example may be arithmetically viewed as modulo 2 binary addition, and as logically equivalent to the exclusive-or (exclusive disjunction) binary logic operation. In this case it can be used for only very simple inputs and outputs, such as 1s and 0s. However, if the number of types of values one can have on the inputs increases, the size of the truth table will increase. For instance, in an addition operation, one needs two operands, A and B. Each can have one of two values, zero or one. The number of combinations of these two values

2544-405: The same term [REDACTED] This disambiguation page lists articles associated with the title Exor . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Exor&oldid=1181443040 " Category : Disambiguation pages Hidden categories: Short description

2597-484: The second, and for each of these there will be two ways of assigning T or F to the third, and so on, giving 2.2.2. …, n times, which is equal to 2 . This results in truth tables like this table "showing that (A→C)∧(B→C) and (A∨B)→C are truth-functionally equivalent ", modeled after a table produced by Howson : If there are n input variables then there are 2 possible combinations of their truth values. A given function may produce true or false for each combination so

2650-440: The summary of Anellis's paper: In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell 's 1912 lecture on "The Philosophy of Logical Atomism" truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes

2703-474: The table For example, the table represents the truth table for Material implication . Logical operators can also be visualized using Venn diagrams . There are 2 nullary operations: The output value is always true, because this operator has zero operands and therefore no input values The output value is never true: that is, always false, because this operator has zero operands and therefore no input values There are 2 unary operations: Logical identity

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2756-424: The use of logic gates or code. For example, a binary addition can be represented with the truth table: where A is the first operand, B is the second operand, C is the carry digit, and R is the result. This truth table is read left to right: This table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values. With respect to

2809-520: The well-known two-element field F 2 {\displaystyle \mathbb {F} _{2}} . This field can represent any logic obtainable with the system ( ∧ , ∨ ) {\displaystyle (\land ,\lor )} and has the added benefit of the arsenal of algebraic analysis tools for fields. More specifically, if one associates F {\displaystyle F} with 0 and T {\displaystyle T} with 1, one can interpret

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