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\(\begin{matrix}
\texttt{(~)}
& = & 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 \(\texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}\) — has the following venn diagram:
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\(\text{Figure 2.} ~~ \texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}\)
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For a contrasting example, the boolean function expressed by the form \(\texttt{((} p \texttt{),(} q \texttt{),(} r \texttt{))}\) has the following venn diagram:
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\(\text{Figure 3.} ~~ \texttt{((} p \texttt{),(} q \texttt{),(} r \texttt{))}\)
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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:
| \(s ~=~ e_1 e_2 \ldots e_{k-1} e_k\),
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\(\begin{array}{llll}
\text{where} & e_j & = & x_j
\\[6pt]
\text{or} & e_j & = & \nu (x_j),
\\[6pt]
\text{for} & j & = & 1 ~\text{to}~ k.
\end{array}\)
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See also
Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-endExternal links
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