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MyWikiBiz, Author Your Legacy — Thursday November 07, 2024
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In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses.  The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation.
 
In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses.  The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation.
   −
: <math>\begin{matrix}
+
{| align="center" cellpadding="10" style="text-align:center"
(~)      & = & 0 & = & \mbox{false} \\
+
|
(x)      & = & \tilde{x} & = & x' \\
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<math>\begin{matrix}
(x, y)    & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\
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(~)      & = & 0 & = & \operatorname{false}
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\\[6pt]
 +
(x)      & = & \tilde{x} & = & x'
 +
\\[6pt]
 +
(x, y)    & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy'
 +
\\[6pt]
 
(x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz'
 
(x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz'
 
\end{matrix}</math>
 
\end{matrix}</math>
 +
|}
    
It may also be noted that <math>(x, y)\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>(x, y)\!</math> and for <math>(x, y, z)\!</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint.  However, the function <math>(x, y, z)\!</math> is not the same thing as the function <math>x + y + z\!</math>.
 
It may also be noted that <math>(x, y)\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>(x, y)\!</math> and for <math>(x, y, z)\!</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint.  However, the function <math>(x, y, z)\!</math> is not the same thing as the function <math>x + y + z\!</math>.
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* A singular proposition ''s''&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B''' can be expressed as a singular conjunction:
 
* A singular proposition ''s''&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B''' can be expressed as a singular conjunction:
<br>
+
 
:{|
+
{| align="center" cellspacing"10" width="90%"
 +
| height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k</math>,
 +
|-
 
|
 
|
| <math>s\ \ =\ e_1 e_2 \ldots e_{k-1} e_k</math>,
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<math>\begin{array}{llll}
|-
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\text{where} & e_j & = & x_j
| where
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\\[6pt]
| <math>e_j\ =\ x_j\!</math>
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\text{or}    & e_j & = & \nu (x_j),
|-
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\\[6pt]
| or
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\text{for}  & j   & = & 1 ~\text{to}~ k.
| <math>e_j\ =\ \nu (x_j)\!</math>,
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\end{array}</math>
|-
  −
| for
  −
| <math>j\ \ =\ 1\ \mbox{to}\ k</math>.
   
|}
 
|}
    
==See also==
 
==See also==
   −
{|
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{{col-begin}}
| valign=top |
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{{col-break}}
 
* [[Ampheck]]
 
* [[Ampheck]]
 
* [[Anamnesis]]
 
* [[Anamnesis]]
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* [[Boolean logic]]
 
* [[Boolean logic]]
 
* [[Boolean-valued function]]
 
* [[Boolean-valued function]]
| valign=top |
+
{{col-break}}
 
* [[Continuous predicate]]
 
* [[Continuous predicate]]
 
* [[Differentiable manifold]]
 
* [[Differentiable manifold]]
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* [[Logical connective]]
 
* [[Logical connective]]
 
* [[Logical graph]]
 
* [[Logical graph]]
| valign=top |
+
{{col-break}}
 
* [[Meno]]
 
* [[Meno]]
 
* [[Multigrade operator]]
 
* [[Multigrade operator]]
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* [[Universal algebra]]
 
* [[Universal algebra]]
 
* [[Zeroth order logic]]
 
* [[Zeroth order logic]]
|}
+
{{col-end}}
    
==External links==
 
==External links==
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