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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' \\ | + | <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>. |
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| | | |
| * A singular proposition ''s'' : '''B'''<sup>''k''</sup> → '''B''' can be expressed as a singular conjunction: | | * A singular proposition ''s'' : '''B'''<sup>''k''</sup> → '''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>,
| + | <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), |
− | |-
| + | \\[6pt] |
− | | 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}} |
− | | valign=top |
| + | {{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== |