Truth table

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A truth table is a tabular array that illustrates the computation of a boolean function, that is, a function of the form $$f : \mathbb{B}^k \to \mathbb{B},$$ where $$k\!$$ is a non-negative integer and $$\mathbb{B}$$ is the boolean domain $$\{ 0, 1 \}.\!$$

Logical negation
Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true.

The truth table of NOT p (also written as ~p or &not;p) is as follows:

The logical negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:

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 of p AND q (also written as p &and; q, p & q, or p$$\cdot$$q) is as follows:

Logical disjunction
Logical disjunction, also called logical alternation, 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 of p OR q (also written as p &or; q) is as follows:

Logical equality
Logical equality 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 of p EQ q (also written as p = q, p &harr; q, or p &equiv; q) is as follows:

Exclusive disjunction
Exclusive disjunction, also known as logical inequality or symmetric difference, is an operation on two logical values, typically the values of two propositions, that produces a value of true just in case exactly one of its operands is true.

The truth table of p XOR q (also written as p + q, p &oplus; q, or p &ne; q) is as follows:

The following equivalents can then be deduced:


 * $$\begin{matrix}

p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ \\     & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ \\     & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}$$

Logical implication
The 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 if and only if 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 &rarr; q) and the logical implication p implies q (symbolized as p &rArr; q) is as follows:

Logical NAND
The logical NAND is a logical 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 of p NAND q (also written as p | q or p &uarr; q) is as follows:

Logical NNOR
The logical NNOR is a logical 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.

The truth table of p NNOR q (also written as p &perp; q or p &darr; q) is as follows:

Translations

 * &#20013;&#25991; : &#30495;&#20540;&#34920;

Focal nodes

 * Inquiry Live


 * Logic Live

Peer nodes

 * Truth Table @ MyWikiBiz
 * Truth Table @ MathWeb Wiki
 * Truth Table @ NetKnowledge
 * Truth Table @ OER Commons


 * Truth Table @ P2P Foundation
 * Truth Table @ SemanticWeb
 * Truth Table @ Subject Wikis
 * Truth Table @ Wikiversity Beta

Logical operators

 * Exclusive disjunction
 * Logical conjunction
 * Logical disjunction
 * Logical equality


 * Logical implication
 * Logical NAND
 * Logical NNOR
 * Negation

Related topics

 * Ampheck
 * Boolean domain
 * Boolean function
 * Boolean-valued function
 * Differential logic


 * Logical graph
 * Minimal negation operator
 * Multigrade operator
 * Parametric operator
 * Peirce's law


 * Propositional calculus
 * Sole sufficient operator
 * Truth table
 * Universe of discourse
 * Zeroth order logic

Relational concepts

 * Continuous predicate
 * Hypostatic abstraction
 * Logic of relatives
 * Logical matrix


 * Relation
 * Relation composition
 * Relation construction
 * Relation reduction


 * Relation theory
 * Relative term
 * Sign relation
 * Triadic relation

Information, Inquiry

 * Inquiry
 * Dynamics of inquiry


 * Semeiotic
 * Logic of information


 * Descriptive science
 * Normative science


 * Pragmatic maxim
 * Truth theory

Related articles

 * Jon Awbrey, &ldquo;Semiotic Information&rdquo;


 * Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;


 * Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;


 * Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;


 * Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;


 * Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;


 * Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;


 * Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;

Document history
Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.


 * Truth Table, MyWikiBiz
 * Truth Table, MathWeb Wiki
 * Truth Table, NetKnowledge
 * Truth Table, OER Commons


 * Truth Table, P2P Foundation
 * Truth Table, SemanticWeb
 * Truth Table, Wikiversity Beta
 * Truth Table, GetWiki


 * Truth Table, Wikinfo
 * Truth Table, Textop Wiki
 * Truth Table, Wikipedia