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| | ''Sheffer bar'' | | ''Sheffer bar'' |
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| − | A [[Logical operation|logical operation]], usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848601.png" />, given by the following [[Truth table|truth table]]:''''''<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/s/s084/s084860/s0848602.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848603.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848604.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">T</td> <td colname="2" style="background-color:white;" colspan="1">T</td> <td colname="3" style="background-color:white;" colspan="1">F</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">T</td> <td colname="2" style="background-color:white;" colspan="1">F</td> <td colname="3" style="background-color:white;" colspan="1">T</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">F</td> <td colname="2" style="background-color:white;" colspan="1">T</td> <td colname="3" style="background-color:white;" colspan="1">T</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">F</td> <td colname="2" style="background-color:white;" colspan="1">F</td> <td colname="3" style="background-color:white;" colspan="1">T</td> </tr> </tbody> </table> | + | A [[Logical operation|logical operation]], usually denoted by $|$, given by the following [[Truth table|truth table]]: |
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| − | </td></tr> </table> | + | <center> |
| | + | {| border="1" class="wikitable" style="text-align:center; width:300px;" |
| | + | |$A$||$B$||$A|B$ |
| | + | |- |
| | + | |$T$||$T$||$F$ |
| | + | |- |
| | + | |$T$||$F$||$T$ |
| | + | |- |
| | + | |$F$||$T$||$T$ |
| | + | |- |
| | + | |$F$||$F$||$T$ |
| | + | |} |
| | + | </center> |
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| − | Thus, the assertion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848605.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848607.png" /> are incompatible, i.e. are not true simultaneously. All other logical operations can be expressed by the Sheffer stroke. For example, the assertion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848608.png" /> (the [[Negation|negation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848609.png" />) is equivalent to the assertion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486010.png" />; the [[Disjunction|disjunction]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486011.png" /> of two assertions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486013.png" /> is expressed as: | + | Thus, the assertion $A|B$ means that $A$ and $B$ are incompatible, i.e. are not true simultaneously. All other logical operations can be expressed by the Sheffer stroke. For example, the assertion $\neg A$ (the [[Negation|negation]] of $A$) is equivalent to the assertion $A|A$; the [[Disjunction|disjunction]] $A\lor B$ of two assertions $A$ and $B$ is expressed as: |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486014.png" /></td> </tr></table>
| + | $$(A|A)|(B|B).$$ |
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| − | The [[Conjunction|conjunction]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486015.png" /> and the [[Implication|implication]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486016.png" /> are expressed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486018.png" />, respectively. Sheffer's stroke was first considered by H. Sheffer. | + | The [[Conjunction|conjunction]] $A\&B$ and the [[Implication|implication]] $A\to B$ are expressed as $(A|B)|(A|B)$ and $A|(B|B)$, respectively. Sheffer's stroke was first considered by H. Sheffer. |
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| | ====References==== | | ====References==== |
Revision as of 12:13, 12 August 2014
Sheffer bar
A logical operation, usually denoted by $|$, given by the following truth table:
| $A$ |
$B$ |
$A|B$
|
| $T$ |
$T$ |
$F$
|
| $T$ |
$F$ |
$T$
|
| $F$ |
$T$ |
$T$
|
| $F$ |
$F$ |
$T$
|
Thus, the assertion $A|B$ means that $A$ and $B$ are incompatible, i.e. are not true simultaneously. All other logical operations can be expressed by the Sheffer stroke. For example, the assertion $\neg A$ (the negation of $A$) is equivalent to the assertion $A|A$; the disjunction $A\lor B$ of two assertions $A$ and $B$ is expressed as:
$$(A|A)|(B|B).$$
The conjunction $A\&B$ and the implication $A\to B$ are expressed as $(A|B)|(A|B)$ and $A|(B|B)$, respectively. Sheffer's stroke was first considered by H. Sheffer.
References
| [1] | H.M. Sheffer, "A set of five independent postulates for Boolean algebras, with applications to logical constants" Trans. Amer. Math. Soc. , 14 (1913) pp. 481–488 |
The Sheffer stroke operation is also called alternative denial.
References
| [a1] | S.C. Kleene, "Introduction to metamathematics" , North-Holland (1950) pp. 139 |
| [a2] | W. Marek, J. Onyszkiewicz, "Elements of logic and the foundations of mathematics in problems" , Reidel & PWN (1982) pp. 4 |
How to Cite This Entry:
Sheffer stroke. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sheffer_stroke&oldid=16985
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article