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A truth table is a mathematical table used in logic — specifically in connection with Boolean algebra, boolean functions, and propositional calculus — to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables. In particular, truth tables can be used to tell whether a propositional expression is true for all legitimate input values, that is, logically valid. Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying variables— to simplify and drastically speed up computation. ...
Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason or principle) is the study of the principles and criteria of valid inference and demonstration. ...
Boolean algebra is the finitary algebra of two values. ...
In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ...
An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ...
In logic, the form of an argument is valid precisely if it cannot lead from true premises to a false conclusion. ...
"The pattern of reasoning that the truth table tabulates was Frege's, Peirce's, and Schröder's by 1880. The tables have been prominent in literature since 1920 (Lukasiewicz, Post, Wittgenstein)" (Quine, 39). Lewis Carroll had formulated truth tables as early as 1894 to solve certain problems, but his manuscripts containing his work on the subject were not discovered until 1977 [1]. Wittgenstein's Tractatus Logico-Philosophicus uses them to place truth functions in a series. The wide influence of this work led to the spread of the use of truth tables. Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ...
Charles Sanders Peirce (IPA: /pÉs/), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ...
Ernst Schröder Ernst Schröder (25 November 1841 Mannheim, Germany - 16 June 1902 Karlsruhe Germany) was a German mathematician mainly known for his work on algebraic logic. ...
The title given to this article is incorrect due to technical limitations. ...
Emil Leon Post (February 11, 1897 - April 21, 1954) was a Polish-American mathematician and logician. ...
Wittgenstein and Hitler in school photograph taken at the Linz Realschule in 1903. ...
W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ...
Charles Lutwidge Dodgson (Lewis Carroll) – believed to be a self-portrait Charles Lutwidge Dodgson (January 27, 1832 – January 14, 1898), better known by the pen name Lewis Carroll, was an English author, mathematician, logician, Anglican clergyman and photographer. ...
Book cover of the Dover edition of Tractatus Logico-Philosophicus (Ogden translation) Tractatus Logico-Philosophicus is the only book-length work published by the philosopher Ludwig Wittgenstein in his lifetime. ...
In logic a truth function is a connective for which the truth value is determined systematically by the values of the statements it connects. ...
Truth tables are used to compute the values of propositional expressions in an effective manner that is sometimes referred to as a decision procedure. A propositional expression is either an atomic formula — a propositional constant, propositional variable, or propositional function term (for example, Px or P(x)) — or built up from atomic formulas by means of logical operators, for example, AND ( ), OR ( ), NOT ( ). For instance,  is a propositional expression. In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ...
In mathematical logic, an atomic formula, or atom, is a formula that has no subformulas. ...
AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ...
OR logic gate. ...
Negation (i. ...
The column headings on a truth table show (i) the propositional functions and/or variables, and (ii) the truth-functional expression built up from those propositional functions or variables and operators. The rows show each possible valuation of T or F assignments to (i) and (ii). In other words, each row is a distinct interpretation of (i) and (ii).
Truth tables for classical logic are limited to Boolean logical systems in which only two logical values are possible, false and true, usually written F and T, or sometimes 0 or 1, respectively. Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ...
Boolean logic is a complete system for logical operations. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
Logical negation
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. Negation, in its most basic sense, changes the truth value of a statement to its opposite. ...
In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...
The truth table for NOT p (also written as ~p or ¬p) is as follows:
Logical Negation | p | ¬p | | F | T | | T | F |
Logical conjunction Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true. AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ...
In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...
The truth table for p AND q (also written as p ∧ q, p & q, or p q) is as follows:
Logical Conjunction | p | q | p ∧ q | | F | F | F | | F | T | F | | T | F | F | | T | T | T | In ordinary language terms, if both p and q are true, then the conjunction p ∧ q is true. For all other assignments of logical values to p and to q the conjunction p ∧ q is false.
It can also be said that if p, then p ∧ q is q, otherwise p ∧ q is p.
Logical disjunction Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false. OR logic gate. ...
In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...
This article does not cite any references or sources. ...
The truth table for p OR q (also written as p ∨ q) is as follows:
Logical Disjunction | p | q | p ∨ q | | F | F | F | | F | T | T | | T | F | T | | T | T | T | Stated in English, if p, then p ∨ q is p, otherwise p ∨ q is q.
Logical implication Logical implication and the material conditional are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false. In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...
In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...
The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:
Logical Implication | p | q | p ⇒ q | | F | F | T | | F | T | T | | T | F | F | | T | T | T |
Logical equality Logical equality (also known as biconditional) is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true. 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 mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...
The truth table for p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:
Logical Equality | p | q | p = q | | F | F | T | | F | T | F | | T | F | F | | T | T | T |
Exclusive disjunction 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. 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 mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...
The truth table for p XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows:
Exclusive Disjunction | p | q | p + q | | F | F | F | | F | T | T | | T | F | T | | T | T | F | For two propositions, XOR can also be written as (p = 1 ∧ q = 0)∨ (p = 0 ∧ q = 1).
Logical NAND The logical NAND is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are true. In other words, it produces a value of true if and only if at least one of its operands is false. NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ...
In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...
The truth table for p NAND q (also written as p | q or p ↑ q) is as follows:
Logical NAND | p | q | p ↑ q | | F | F | T | | F | T | T | | T | F | T | | T | T | F | It is frequently useful to express a logical operation as a compound operation, that is, as an operation that is built up or composed from other operations. Many such compositions are possible, depending on the operations that are taken as basic or "primitive" and the operations that are taken as composite or "derivative".
In the case of logical NAND, it is clearly expressible as a compound of NOT and AND.
The negation of conjunction  , and the disjunction of negations  are depicted as follows:
| p | q |  |  |  |  | | | F | F | F | T | T | T | T | | F | T | F | T | T | F | T | | T | F | F | T | F | T | T | | T | T | T | F | F | F | F |
Logical NOR The logical NOR is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are false. In other words, it produces a value of false if and only if at least one of its operands is true. ↓ is also known as the Peirce arrow after its inventor, Charles Peirce, and is a Sole sufficient operator. NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ...
In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...
NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ...
Charles Sanders Peirce (IPA: /pÉs/), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ...
Sole sufficient operator, or sole sufficient connective, is a term used to describe an operator that is sufficient by itself to generate all of the operators in a specified class of operators. ...
The truth table for p NOR q (also written as p ⊥ q or p ↓ q) is as follows:
Logical NOR | p | q | p ↓ q | | F | F | T | | F | T | F | | T | F | F | | T | T | F | The negation of disjunction  and the conjunction of negations  are tabulated as follows: | p | q |  |  | | | | | F | F | F | T | T | T | T | | F | T | T | F | T | F | F | | T | F | T | F | F | T | F | | T | T | T | F | F | F | F | Inspection of the tabular derivations for NAND and NOR, under each assignment of logical values to the functional arguments  and  , produces the identical patterns of functional values for as for , and for as for . Thus the first and second expressions in each pair are logically equivalent, and may be substituted for each other in all contexts that pertaing solely to their logical values.
This equivalence is one of De Morgan's laws. note that demorgans laws are also a big part in circut design. ...
Applications Truth tables can be used to prove many other logical equivalences. For example, consider the following truth table: In logic, statements p and q are logically equivalent if they have the same logical content. ...
Logical Equivalence : (p → q) = (¬p ∨ q) | p | q | ¬p | ¬p ∨ q | p → q | | F | F | T | T | T | | F | T | T | T | T | | T | F | F | F | F | | T | T | F | T | T | This demonstrates the fact that p → q is logically equivalent to ¬p ∨ q. In logic, statements p and q are logically equivalent if they have the same logical content. ...
Truth table for most commonly used logical operators Here is a truth table giving definitions of the most commonly used 6 of the 16 possible truth functions of 2 binary variables (P,Q are thus boolean variables): Book cover of the Dover edition of Tractatus Logico-Philosophicus (Ogden translation) Tractatus Logico-Philosophicus is the only book-length work published by the philosopher Ludwig Wittgenstein in his lifetime. ...
| P | Q |  |  |  |  |  |  | | F | F | F | F | F | T | T | T | | F | T | F | T | T | F | T | F | | T | F | F | T | T | F | F | T | | T | T | T | T | F | T | T | T | Key: - T = true, F = false
- = AND (logical conjunction)
- = OR (logical disjunction)
 = XOR (exclusive or) = XNOR (exclusive nor) = conditional "if-then" = conditional "(then)-if"  biconditional or "if-and-only-if" is logically equivalent to : XNOR (exclusive nor). Johnston diagrams, similar to Venn diagrams and Euler diagrams, provide a way of visualizing truth tables. An interactive Johnston diagram illustrating truth tables is at LogicTutorial.com AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ...
OR logic gate. ...
Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ...
XNOR Logic Gate Symbol Exclusive nor (usual symbol XNOR occasionally XAND <exclusive and>) is a logical operator in Boolean algebra. ...
In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...
In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...
This article does not cite any references or sources. ...
In logic, statements p and q are logically equivalent if they have the same logical content. ...
XNOR Logic Gate Symbol Exclusive nor (usual symbol XNOR occasionally XAND <exclusive and>) is a logical operator in Boolean algebra. ...
Johnston diagrams, which look similar to Euler or Venn diagrams, illustrate formal propositional logic in a visual manner. ...
A Venn diagram of sets A, B, and C Venn diagrams are illustrations used in the branch of mathematics known as set theory. ...
A Venn diagram of sets A, B, and C Venn diagrams are illustrations used in the branch of mathematics known as set theory. ...
Johnston diagrams, which look similar to Euler or Venn diagrams, illustrate formal propositional logic in a visual manner. ...
Condensed truth tables for binary operators 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: Boolean logic is a complete system for logical operations. ...
This notation 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 the distribution of the values in the table which can assist the reader in grasping the rules more quickly.
Truth tables in digital logic Truth tables are also used to specify the functionality 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 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. In computer science, a lookup table is a data structure, usually an array or associative array, used to replace a runtime computation with a simpler lookup operation. ...
Digital circuits are electric circuits based on a number of discrete voltage levels. ...
This article is about the unit of information. ...
The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ...
The integers are commonly denoted by the above symbol. ...
PCB Layout Program Electronic design automation (EDA) is the category of tools for designing and producing electronic systems ranging from printed circuit boards (PCBs) to integrated circuits. ...
Computer software (or simply software) refers to one or more computer programs and data held in the storage of a computer for some purpose. ...
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 the LUT's output value is the kth 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 ith input is true, let Vi = 1, else let Vi = 0. Then the kth bit of the binary representation of the truth table is the LUT's output value, where k = V0*2^0 + V1*2^1 + V2*2^2 + ... + Vn*2^n. This article does not cite any references or sources. ...
Truth tables are 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 mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ...
A binary decision diagram (BDD), like a negation normal form (NNF) or a propositional directed acyclic graph (PDAG), is a data structure that is used to represent a Boolean function. ...
Applications of truth tables in digital electronics In digital electronics (and computer science, fields of engineering derived from applied logic and math), truth tables can be used to reduce basic boolean operations to simple correlations of inputs to outputs, without the use of logic gates or code. For example, a binary addition can be represented with the truth table: A B | C R 1 1 | 1 0 1 0 | 0 1 0 1 | 0 1 0 0 | 0 0 where A = First Operand B = Second Operand C = Carry R = Result This truth table is read left to right: - Value pair (A,B) equals value pair (C,R).
- Or for this example, A plus B equal result R, with the Carry C.
Note that this table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values.
In this case it can only be used for very simple inputs and outputs, such as 1's and 0's, 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 is 2x2, or four. So the result is four possible outputs of C and R. If one was to use base 3, the size would increase to 3x3, 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: In electronics, an adder is a device which will perform the addition, S, of two numbers. ...
A B C* | C R 0 0 0 | 0 0 0 1 0 | 0 1 1 0 0 | 0 1 1 1 0 | 1 0 0 0 1 | 0 1 0 1 1 | 1 0 1 0 1 | 1 0 1 1 1 | 1 1 Same as previous, but.. C* = Carry from previous adder
References - Quine, W.V. (1982), Methods of Logic, 4th edition, Harvard University Press, Cambridge, MA.
W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ...
See also
Basic logical operators 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. ...
AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of 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 logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...
NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ...
NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ...
Negation (i. ...
Related topics Ampheck, from Greek double-edged, is a term coined by Charles Sanders Peirce for either one of the pair of logically dual operators, variously referred to as Peirce arrows, Sheffer strokes, or NAND and NNOR. Either of these logical operators is a sole sufficient operator for deriving or generating all...
Boolean algebra is the finitary algebra of two values. ...
Algebra of sets Ampheck Boole, George Boolean algebra Boolean domain Boolean function Boolean logic Boolean implicant Boolean prime ideal theorem Boolean-valued function Boolean-valued model Boolean satisfiability problem Booles syllogistic Canonical form (Boolean algebra) Characteristic function Compactness theorem Complete Boolean algebra De Morgan, Augustus De Morgans laws...
A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. ...
In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
A boolean-valued function, in some usages a predicate or a proposition, is a function of the type f : X → B, where X is an arbitrary set and where B is a boolean domain. ...
First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...
The Karnaugh map, also known as a Veitch diagram (K-map or KV-map for short), is a tool to facilitate management of Boolean algebraic expressions. ...
In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ...
A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
In logic and mathematics, the minimal negation operator is a multigrade operator where each is a k-ary boolean function defined in such a way that if and only if exactly one of the arguments is 0. ...
In logic and mathematics, a multigrade operator is a parametric operator with parameter k in the set N of non-negative integers. ...
In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ...
In logic and mathematics, a parametric operator with parameter in the parametric set is a indexed family of operators with index in the index set . ...
In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ...
Sole sufficient operator, or sole sufficient connective, is a term used to describe an operator that is sufficient by itself to generate all of the operators in a specified class of operators. ...
Zeroth-order logic is a term in popular use among practitioners for the subject matter otherwise known as boolean functions, monadic predicate logic, propositional calculus, or sentential calculus. ...
External links - Web-based truth table generator
- Boolean expression evaluator, generates truth table (Java applet)
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