|
\begin{matrix}
\texttt{()}
& = & \nu_0
& = & 0
& = & \operatorname{false}
\'"`UNIQ-MathJax1-QINU`"'
* The point \((0, 0, \ldots , 0, 0) with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to 1,\! namely, the point where:
- (x_1)(x_2)\ldots(x_{n-1})(x_n) = 1.
To pass from these limiting examples to the general case, observe that a singular proposition s : \mathbb{B}^k \to \mathbb{B} can be given canonical expression as a conjunction of literals, s = e_1 e_2 \ldots e_{k-1} e_k. Then the proposition \nu (e_1, e_2, \ldots, e_{k-1}, e_k) is 1\! on the points adjacent to the point where s\! is 1,\! and 0 everywhere else on the cube.
For example, consider the case where k = 3.\! Then the minimal negation operation \nu (p, q, r)\! — written more simply as \texttt{(p, q, r)} — has the following venn diagram:
|
\text{Figure 2.}~~\texttt{(p, q, r)}
|
For a contrasting example, the boolean function expressed by the form \texttt{((p),(q),(r))} has the following venn diagram:
|
\text{Figure 3.}~~\texttt{((p),(q),(r))}
|
Glossary of basic terms
- Boolean domain
- A boolean domain \mathbb{B} is a generic 2-element set, for example, \mathbb{B} = \{ 0, 1 \}, whose elements are interpreted as logical values, usually but not invariably with 0 = \operatorname{false} and 1 = \operatorname{true}.
- Boolean variable
- A boolean variable x\! is a variable that takes its value from a boolean domain, as x \in \mathbb{B}.
- Proposition
- In situations where boolean values are interpreted as logical values, a boolean-valued function f : X \to \mathbb{B} or a boolean function g : \mathbb{B}^k \to \mathbb{B} is frequently called a proposition.
- Basis element, Coordinate projection
- Given a sequence of k\! boolean variables, x_1, \ldots, x_k, each variable x_j\! may be treated either as a basis element of the space \mathbb{B}^k or as a coordinate projection x_j : \mathbb{B}^k \to \mathbb{B}.
- Basic proposition
- This means that the set of objects \{ x_j : 1 \le j \le k \} is a set of boolean functions \{ x_j : \mathbb{B}^k \to \mathbb{B} \} subject to logical interpretation as a set of basic propositions that collectively generate the complete set of 2^{2^k} propositions over \mathbb{B}^k.
- Literal
- A literal is one of the 2k\! propositions x_1, \ldots, x_k, (x_1), \ldots, (x_k), in other words, either a posited basic proposition x_j\! or a negated basic proposition (x_j),\! for some j = 1 ~\text{to}~ k.
- Fiber
- In mathematics generally, the fiber of a point y \in Y under a function f : X \to Y is defined as the inverse image f^{-1}(y) \subseteq X.
- In the case of a boolean function f : \mathbb{B}^k \to \mathbb{B}, there are just two fibers:
- The fiber of 0\! under f,\! defined as f^{-1}(0),\! is the set of points where the value of f\! is 0.\!
- The fiber of 1\! under f,\! defined as f^{-1}(1),\! is the set of points where the value of f\! is 1.\!
- Fiber of truth
- When 1\! is interpreted as the logical value \operatorname{true}, then f^{-1}(1)\! is called the fiber of truth in the proposition f.\! Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation [|f|] = f^{-1}(1)\! for the fiber of truth in the proposition f.\!
- Singular boolean function
- A singular boolean function s : \mathbb{B}^k \to \mathbb{B} is a boolean function whose fiber of 1\! is a single point of \mathbb{B}^k.
- Singular proposition
- In the interpretation where 1\! equals \operatorname{true}, a singular boolean function is called a singular proposition.
- Singular boolean functions and singular propositions serve as functional or logical representatives of the points in \mathbb{B}^k.
- Singular conjunction
- A singular conjunction in \mathbb{B}^k \to \mathbb{B} is a conjunction of k\! literals that includes just one conjunct of the pair \{ x_j, ~\nu(x_j) \} for each j = 1 ~\text{to}~ k.
- A singular proposition s : \mathbb{B}^k \to \mathbb{B} can be expressed as a singular conjunction:
| s ~=~ e_1 e_2 \ldots e_{k-1} e_k,
|
|
\begin{array}{llll}
\text{where} & e_j & = & x_j
\\[6pt]
\text{or} & e_j & = & \nu (x_j),
\\[6pt]
\text{for} & j & = & 1 ~\text{to}~ k.
\end{array}
|
See also
Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-endExternal links
<sharethis />
|
.jpg)
,(Q),(R)).jpg)
