Difference between revisions of "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 | + | A '''truth table''' is a tabular array that illustrates the computation of a [[boolean function]], that is, a function of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{B}</math> is the [[boolean domain]] <math>\{ 0, 1 \}.\!</math> |
==Logical negation== | ==Logical negation== | ||
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The truth table of '''NOT p''' (also written as '''~p''' or '''¬p''') is as follows: | The truth table of '''NOT p''' (also written as '''~p''' or '''¬p''') is as follows: | ||
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background: | + | <br> |
| + | |||
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:40%" | ||
|+ '''Logical Negation''' | |+ '''Logical Negation''' | ||
| − | |- style="background: | + | |- style="background:#e6e6ff" |
! style="width:20%" | p | ! style="width:20%" | p | ||
! style="width:20%" | ¬p | ! style="width:20%" | ¬p | ||
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| T || F | | T || F | ||
|} | |} | ||
| + | |||
<br> | <br> | ||
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: | 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: | ||
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background: | + | <br> |
| + | |||
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; width:40%" | ||
|+ '''Variant Notations''' | |+ '''Variant Notations''' | ||
| − | |- style="background: | + | |- style="background:#e6e6ff" |
! style="text-align:center" | Notation | ! style="text-align:center" | Notation | ||
! Vocalization | ! Vocalization | ||
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| bang ''p'' | | bang ''p'' | ||
|} | |} | ||
| + | |||
<br> | <br> | ||
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The truth table of '''p AND q''' (also written as '''p ∧ q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows: | The truth table of '''p AND q''' (also written as '''p ∧ q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows: | ||
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background: | + | <br> |
| + | |||
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" | ||
|+ '''Logical Conjunction''' | |+ '''Logical Conjunction''' | ||
| − | |- style="background: | + | |- style="background:#e6e6ff" |
! style="width:15%" | p | ! style="width:15%" | p | ||
! style="width:15%" | q | ! style="width:15%" | q | ||
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| T || T || T | | T || T || T | ||
|} | |} | ||
| + | |||
<br> | <br> | ||
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The truth table of '''p OR q''' (also written as '''p ∨ q''') is as follows: | The truth table of '''p OR q''' (also written as '''p ∨ q''') is as follows: | ||
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background: | + | <br> |
| + | |||
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" | ||
|+ '''Logical Disjunction''' | |+ '''Logical Disjunction''' | ||
| − | |- style="background: | + | |- style="background:#e6e6ff" |
! style="width:15%" | p | ! style="width:15%" | p | ||
! style="width:15%" | q | ! style="width:15%" | q | ||
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| T || T || T | | T || T || T | ||
|} | |} | ||
| + | |||
<br> | <br> | ||
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The truth table of '''p EQ q''' (also written as '''p = q''', '''p ↔ q''', or '''p ≡ q''') is as follows: | The truth table of '''p EQ q''' (also written as '''p = q''', '''p ↔ q''', or '''p ≡ q''') is as follows: | ||
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background: | + | <br> |
| + | |||
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" | ||
|+ '''Logical Equality''' | |+ '''Logical Equality''' | ||
| − | |- style="background: | + | |- style="background:#e6e6ff" |
! style="width:15%" | p | ! style="width:15%" | p | ||
! style="width:15%" | q | ! style="width:15%" | q | ||
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| T || T || T | | T || T || T | ||
|} | |} | ||
| + | |||
<br> | <br> | ||
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The truth table of '''p XOR q''' (also written as '''p + q''', '''p ⊕ q''', or '''p ≠ q''') is as follows: | The truth table of '''p XOR q''' (also written as '''p + q''', '''p ⊕ q''', or '''p ≠ q''') is as follows: | ||
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background: | + | <br> |
| + | |||
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" | ||
|+ '''Exclusive Disjunction''' | |+ '''Exclusive Disjunction''' | ||
| − | |- style="background: | + | |- style="background:#e6e6ff" |
! style="width:15%" | p | ! style="width:15%" | p | ||
! style="width:15%" | q | ! style="width:15%" | q | ||
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| T || T || F | | T || T || F | ||
|} | |} | ||
| + | |||
<br> | <br> | ||
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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 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: | ||
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background: | + | <br> |
| + | |||
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" | ||
|+ '''Logical Implication''' | |+ '''Logical Implication''' | ||
| − | |- style="background: | + | |- style="background:#e6e6ff" |
! style="width:15%" | p | ! style="width:15%" | p | ||
! style="width:15%" | q | ! style="width:15%" | q | ||
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| T || T || T | | T || T || T | ||
|} | |} | ||
| + | |||
<br> | <br> | ||
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The truth table of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows: | The truth table of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows: | ||
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background: | + | <br> |
| + | |||
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" | ||
|+ '''Logical NAND''' | |+ '''Logical NAND''' | ||
| − | |- style="background: | + | |- style="background:#e6e6ff" |
! style="width:15%" | p | ! style="width:15%" | p | ||
! style="width:15%" | q | ! style="width:15%" | q | ||
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| T || T || F | | T || T || F | ||
|} | |} | ||
| + | |||
<br> | <br> | ||
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The truth table of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: | The truth table of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: | ||
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background: | + | <br> |
| + | |||
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" | ||
|+ '''Logical NNOR''' | |+ '''Logical NNOR''' | ||
| − | |- style="background: | + | |- style="background:#e6e6ff" |
! style="width:15%" | p | ! style="width:15%" | p | ||
! style="width:15%" | q | ! style="width:15%" | q | ||
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| T || T || F | | T || T || F | ||
|} | |} | ||
| + | |||
<br> | <br> | ||
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[[Category:Computer Science]] | [[Category:Computer Science]] | ||
[[Category:Discrete Mathematics]] | [[Category:Discrete Mathematics]] | ||
| + | [[Category:Formal Languages]] | ||
| + | [[Category:Formal Sciences]] | ||
| + | [[Category:Formal Systems]] | ||
| + | [[Category:Linguistics]] | ||
[[Category:Logic]] | [[Category:Logic]] | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
| + | [[Category:Philosophy]] | ||
| + | [[Category:Semiotics]] | ||
| + | |||
| + | <sharethis /> | ||
Revision as of 15:10, 26 May 2009
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 |
