Changes

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' \\

<math>\begin{matrix}

(x, y)    & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\

(~)      & = & 0 & = & \operatorname{false}

\\[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>.

* A singular proposition ''s''&nbsp;:&nbsp;'''B'''^''k''&nbsp;→&nbsp;'''B''' can be expressed as a singular conjunction:

* A singular proposition ''s''&nbsp;:&nbsp;'''B'''^''k''&nbsp;→&nbsp;'''B''' can be expressed as a singular conjunction:

<br>

 

:{|

| 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>,

<math>\begin{array}{llll}

\text{where} & e_j & = & x_j

| where

\\[6pt]

| <math>e_j\ =\ x_j\!</math>

\text{or}    & e_j & = & \nu (x_j),

| or

\text{for}  & j   & = & 1 ~\text{to}~ k.

| <math>e_j\ =\ \nu (x_j)\!</math>,

\end{array}</math>

| for

| <math>j\ \ =\ 1\ \mbox{to}\ k</math>.

|}

|}

==See also==

==See also==

{|

{{col-begin}}

* [[Ampheck]]

* [[Ampheck]]

* [[Anamnesis]]

* [[Anamnesis]]

* [[Boolean logic]]

* [[Boolean logic]]

* [[Boolean-valued function]]

* [[Boolean-valued function]]

* [[Continuous predicate]]

* [[Continuous predicate]]

* [[Differentiable manifold]]

* [[Differentiable manifold]]

* [[Logical connective]]

* [[Logical connective]]

* [[Logical graph]]

* [[Logical graph]]

* [[Meno]]

* [[Meno]]

* [[Multigrade operator]]

* [[Multigrade operator]]

* [[Universal algebra]]

* [[Universal algebra]]

* [[Zeroth order logic]]

* [[Zeroth order logic]]

|}

{{col-end}}

==External links==

==External links==