truth table
Information about truth table
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.
"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.
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.
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.
The truthling table for NOT p (also written as ~p or ¬p) is as follows:
The truth table for p AND q (also written as p ∧ q, p & q, or p
q) is as follows:
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.
The truth table for p OR q (also written as p ∨ q) is as follows:
Stated in English, if p, then p ∨ q is p, otherwise p ∨ q is q.
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:
The truth table for p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:
The truth table for p XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows:
For two propositions, XOR can also be written as (p = 1 ∧ q = 0)∨ (p = 0 ∧ q = 1).
The truth table for p NAND q (also written as p | q or p ↑ q) is as follows:
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:
The truth table for p NOR q (also written as p ⊥ q or p ↓ q) is as follows:
The negation of disjunction
and the conjunction of negations 
are tabulated as follows:
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.
This demonstrates the fact that p → q is logically equivalent to ¬p ∨ q.
Key:
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
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.
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.
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.
This truth table is read left to right:
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:
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Western Philosophy
19th/20th century philosophy
Name: Charles Sanders Peirce
Birth: September 10, 1839
Cambridge, Massachusetts
Death: April 19, 1914
Milford, Pennsylvania
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In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true.
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In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true.
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"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.
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.
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.
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.The truthling table for NOT p (also written as ~p or ¬p) is as follows:
| 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.The truth table for p AND q (also written as p ∧ q, p & q, or p
q) is as follows:
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
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.The truth table for p OR q (also written as p ∨ q) is as follows:
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
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 the singular case the first operand is true and the second operand is false.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:
| p | q | p ⇒ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | 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.The truth table for p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:
| p | q | p ≡ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | 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.The truth table for p XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows:
| p | q | p + q |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | 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.The truth table for p NAND q (also written as p | q or p ↑ q) is as follows:
| p | q | p ↑ q |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
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:
 |  |  | |  |  | |
|---|---|---|---|---|---|---|
| T | T | T | F | F | F | F |
| T | F | F | T | F | T | T |
| F | T | F | T | T | F | T | F || F || F || T || T || T || T |
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.The truth table for p NOR q (also written as p ⊥ q or p ↓ q) is as follows:
| p | q | p ↓ q |
|---|---|---|
| T | T | F |
| T | F | F |
| F | T | F |
| F | F | T |
The negation of disjunction
and the conjunction of negations are tabulated as follows:
|
|
 |
|
|
|
|
|---|---|---|---|---|---|---|
| T | T | T | F | F | F | F |
| T | F | T | F | F | T | F |
| F | T | T | F | T | F | F | F || F || F || T || T || T || T |
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.
Applications
Truth tables can be used to prove many other logical equivalences. For example, consider the following truth table:| 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.
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): |  |  |  |  |  |  |  |
|---|---|---|---|---|---|---|---|
| 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)
= 
= 
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
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:
|
|
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.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.
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.
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.
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:
> 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.
See also
Basic logical operators
Related topics
External links
Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation.
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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.
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A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. These play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers.
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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
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expression must be well-formed. That is, the operators must have the correct number of inputs, in the correct places. The expression 2 + 3 is well formed; the expression * 2 + is not, at least, not in the usual notation of arithmetic.
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validity as it occurs in logic refers generally to a property of deductive arguments, although many logic texts apply the term to statements as well (a statement is a sentence that “has a truth value,” i.e., that is either true or false).
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Friedrich Ludwig Gottlob Frege
Birth: November 8, 1848
Death: 26 July, 1925
School/tradition: Analytic philosophy
Main interests: Philosophy of mathematics, mathematical logic, Philosophy of language
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Birth: November 8, 1848
Death: 26 July, 1925
School/tradition: Analytic philosophy
Main interests: Philosophy of mathematics, mathematical logic, Philosophy of language
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ierce]].
Western Philosophy
19th/20th century philosophy
Name: Charles Sanders Peirce
Birth: September 10, 1839
Cambridge, Massachusetts
Death: April 19, 1914
Milford, Pennsylvania
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Ernst Schröder (25 November, 1841 Mannheim, Germany – 16 June, 1902 Karlsruhe Germany) was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic (a term he may have invented), by virtue of summarizing
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Emil Leon Post
Born February 11, 1897
Augustów, then Russian Empire ,
today Poland
Died April 21 1954,
New York City, U.S.
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Born February 11, 1897
Augustów, then Russian Empire ,
today Poland
Died April 21 1954,
New York City, U.S.
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Ludwig Josef Johann Wittgenstein (IPA: ['luːtvɪç 'joːzɛf 'joːhan 'vɪtgənʃtaɪn]
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Willard Van Orman Quine (June 25, 1908 – December 25, 2000), usually cited as W.V. Quine or W.V.O. Quine but known to his friends as Van, was one of the most influential philosophers and logicians of the 20th century.
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Charles Lutwidge Dodgson (IPA: /ˈdɒdsən/) (January 27 1832 – January 14 1898), better known by the pen name Lewis Carroll (/ˈkærəl/
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Tractatus Logico-Philosophicus is the only book-length work published by the philosopher Ludwig Wittgenstein in his lifetime. It was written while he was a soldier on leave during World War I in 1918.
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truth function is a function from a set of truth-values to truth-values. Classically the domain and range of a truth function are , but generally they may have any number of truth-values, including an infinity of them.
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In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem.
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In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula with no logical connectives or strict sub-formulas. Atoms are thus the simplest well-formed formulas of the logic.
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In logic and/or mathematics, logical conjunction or 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!
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Definition
Logical conjunction..... Click the link for more information.
or, also known as logical disjunction or inclusive disjunction is a logical operator that results in true whenever one or more of its operands are true. In grammar, or is a coordinating conjunction.
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For other uses, see .
In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true.
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Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. They are characterised by a number of properties[1]; non-classical logics are those that lack one or more of these properties, which are:
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In logic and mathematics, a logical value, also called a truth value, is a value indicating the extent to which a proposition is true.
In classical logic, the only possible truth values are true and false.
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In classical logic, the only possible truth values are true and false.
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For other uses, see .
In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true.
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In logic and mathematics, a logical value, also called a truth value, is a value indicating the extent to which a proposition is true.
In classical logic, the only possible truth values are true and false.
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In classical logic, the only possible truth values are true and false.
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proposition is the content of an assertion, that is, it is true-or-false and defined by the meaning of a particular piece of language. The proposition is independent of the of communication.
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In logic and/or mathematics, logical conjunction or 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!
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Definition
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