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==Logical negation==

==Logical negation==

''[[Logical negation]]'' is an [[logical operation|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.

'''[[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:

The [[truth table]] of <math>\operatorname{NOT}~ p,</math> also written <math>\lnot p,\!</math> appears below:

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:40%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"

|+ '''Logical Negation'''

|+ style="height:30px" | <math>\text{Logical Negation}\!</math>

|- style="background:#e6e6ff"

|- style="height:40px; background:#f0f0ff"

! style="width:20%" | p

| style="width:50%" | <math>p\!</math>

! style="width:20%" | &not;p

| style="width:50%" | <math>\lnot p\!</math>

|-

|-

| F || T

| <math>\operatorname{F}</math> || <math>\operatorname{T}</math>

|-

|-

| T || F

| <math>\operatorname{T}</math> || <math>\operatorname{F}</math>

|}

|}

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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 negation of a proposition <math>p\!</math> may be found notated in various ways in various contexts of application, often merely for typographical convenience.  Among these variants are the following:

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; width:40%"

{| align="center" border="1" cellpadding="8" cellspacing="0" width="45%"

|+ '''Variant Notations'''

|+ style="height:30px" | <math>\text{Variant Notations}\!</math>

|- style="background:#e6e6ff"

|- style="height:40px; background:#f0f0ff"

! style="text-align:center" | Notation

| width="50%" align="center" | <math>\text{Notation}\!</math>

! Vocalization

| width="50%" | <math>\text{Vocalization}\!</math>

|-

|-

| style="text-align:center" | <math>\bar{p}</math>

| align="center" | <math>\bar{p}\!</math>

| bar ''p''

| <math>p\!</math> bar

|-

|-

| style="text-align:center" | <math>p'\!</math>

| align="center" | <math>\tilde{p}\!</math>

| ''p'' prime,<p> ''p'' complement

| <math>p\!</math> tilde

|-

|-

| style="text-align:center" | <math>!p\!</math>

| align="center" | <math>p'\!</math>

| bang ''p''

|-

| align="center" | <math>!p\!</math>

| bang <math>p\!</math>

|}

|}