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| − | The logical operation as a result of which, for a given statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n0661701.png" />, the statement "not A" is obtained. In formal languages, the statement obtained as result of the negation of a statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n0661702.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n0661703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n0661704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n0661705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n0661706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n0661707.png" /> (these are read: "not A" , "it is not true that A" , "A does not hold" , etc.). Semantically, the negation of a statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n0661708.png" /> signifies that the assumption <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n0661709.png" /> leads to a contradiction (cf. [[Contradiction (inconsistency)|Contradiction (inconsistency)]]). In classical two-valued logic the following [[Truth table|truth table]] applies for the operation of negation:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n06617010.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n06617011.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n06617012.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n06617013.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n06617014.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066170/n06617015.png" /></td> </tr> </tbody> </table> | + | {{TEX|done}} |
| | + | The logical operation as a result of which, for a given statement $A$, the statement "not A" is obtained. In formal languages, the statement obtained as result of the negation of a statement $A$ is denoted by $\neg A$, $\sim A$, $\overline A$, $-A$, $A'$ (these are read: "not A", "it is not true that A", "A does not hold", etc.). Semantically, the negation of a statement $A$ signifies that the assumption $A$ leads to a contradiction (cf. [[Contradiction (inconsistency)|Contradiction (inconsistency)]]). In classical two-valued logic the following [[Truth table|truth table]] applies for the operation of negation: |
| | | | |
| − | </td></tr> </table> | + | <center> |
| | + | {| border="1" class="wikitable" style="text-align:center; width:300px;" |
| | + | |$A$||$\neg A$ |
| | + | |- |
| | + | |$T$||$F$ |
| | + | |- |
| | + | |$F$||$T$ |
| | + | |} |
| | + | </center> |
Latest revision as of 07:34, 12 August 2014
The logical operation as a result of which, for a given statement $A$, the statement "not A" is obtained. In formal languages, the statement obtained as result of the negation of a statement $A$ is denoted by $\neg A$, $\sim A$, $\overline A$, $-A$, $A'$ (these are read: "not A", "it is not true that A", "A does not hold", etc.). Semantically, the negation of a statement $A$ signifies that the assumption $A$ leads to a contradiction (cf. Contradiction (inconsistency)). In classical two-valued logic the following truth table applies for the operation of negation:
| $A$ |
$\neg A$
|
| $T$ |
$F$
|
| $F$ |
$T$
|
How to Cite This Entry:
Negation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negation&oldid=18029
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article