Difference between revisions of "Truth table"
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* [[Boolean function]] | * [[Boolean function]] | ||
* [[Boolean-valued function]] | * [[Boolean-valued function]] | ||
| + | * [[Differential logic]] | ||
{{col-break}} | {{col-break}} | ||
* [[Logical graph]] | * [[Logical graph]] | ||
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* [[Minimal negation operator]] | * [[Minimal negation operator]] | ||
| + | * [[Multigrade operator]] | ||
| + | * [[Parametric operator]] | ||
* [[Peirce's law]] | * [[Peirce's law]] | ||
{{col-break}} | {{col-break}} | ||
* [[Propositional calculus]] | * [[Propositional calculus]] | ||
| + | * [[Sole sufficient operator]] | ||
* [[Truth table]] | * [[Truth table]] | ||
* [[Universe of discourse]] | * [[Universe of discourse]] | ||
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{{col-begin}} | {{col-begin}} | ||
{{col-break}} | {{col-break}} | ||
| + | * [[Continuous predicate]] | ||
| + | * [[Hypostatic abstraction]] | ||
* [[Logic of relatives]] | * [[Logic of relatives]] | ||
| + | * [[Logical matrix]] | ||
| + | {{col-break}} | ||
* [[Relation (mathematics)|Relation]] | * [[Relation (mathematics)|Relation]] | ||
* [[Relation composition]] | * [[Relation composition]] | ||
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* [[Relation construction]] | * [[Relation construction]] | ||
* [[Relation reduction]] | * [[Relation reduction]] | ||
| + | {{col-break}} | ||
* [[Relation theory]] | * [[Relation theory]] | ||
| − | |||
* [[Relative term]] | * [[Relative term]] | ||
* [[Sign relation]] | * [[Sign relation]] | ||
Revision as of 14:18, 29 April 2010
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 ¬p) is as follows:
| p | ¬p |
|---|---|
| F | T |
| T | F |
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:
| Notation | Vocalization |
|---|---|
| \(\bar{p}\) | bar p |
| \(p'\!\) | p prime, p complement |
| \(!p\!\) | bang p |
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 ∧ q, p & q, or p\(\cdot\)q) is as follows:
| p | q | p ∧ q |
|---|---|---|
| F | F | F |
| F | T | F |
| T | F | F |
| T | T | T |
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 ∨ q) is as follows:
| p | q | p ∨ q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
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 ↔ q, or p ≡ q) is as follows:
| p | q | p = q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | T |
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 ⊕ q, or p ≠ q) is as follows:
| p | q | p XOR q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | F |
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 → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:
| p | q | p ⇒ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
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 ↑ q) is as follows:
| p | q | p ↑ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | T |
| T | T | F |
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 ⊥ q or p ↓ q) is as follows:
| p | q | p ↓ q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | F |
Syllabus
Focal nodes
Template:Col-breakTemplate:Col-breakTemplate:Col-endPeer nodes
Logical operators
Related topics
- Propositional calculus
- Sole sufficient operator
- Truth table
- Universe of discourse
- Zeroth order logic
Relational concepts
Related articles
Translations
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.
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