Changes

==Charts and graphs==

==Charts and graphs==

Two ways of visualizing the space <math>\mathbb{B}^k</math> of <math>2^k\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture.  The hypercube picture associates each point of <math>\mathbb{B}^k</math> with a corresponding point of the <math>k\!</math>-cube.  The venn diagram picture associates each point of <math>\mathbb{B}^k</math> with a corresponding "cell" of the venn diagram on <math>k\!</math> "circles".

Two ways of visualizing the space <math>\mathbb{B}^k</math> of <math>2^k\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture.  The hypercube picture associates each point in <math>\mathbb{B}^k</math> with a unique point in the <math>k\!</math>-dimensional hypercube.  The venn diagram picture associates each point in <math>\mathbb{B}^k</math> with a unique "cell" in the venn diagram on <math>k\!</math> "circles".

In addition, each point of <math>\mathbb{B}^k</math> is the unique point in the '''[[fiber (mathematics)|fiber]] of truth''' <math>[|s|]\!</math> of a '''singular proposition''' <math>s : \mathbb{B}^k \to \mathbb{B},</math> and thus it is the unique point where a '''singular conjunction''' of <math>k\!</math> '''literals''' is equal to 1.

In addition, each point of <math>\mathbb{B}^k</math> is the unique point in the '''[[fiber (mathematics)|fiber]] of truth''' <math>[|s|]\!</math> of a '''singular proposition''' <math>s : \mathbb{B}^k \to \mathbb{B},</math> and thus it is the unique point where a '''singular conjunction''' of <math>k\!</math> '''literals''' is equal to 1.

:: <math>(x_1)(x_2)\ldots(x_{n-1})(x_n) = 1.</math>

:: <math>(x_1)(x_2)\ldots(x_{n-1})(x_n) = 1.</math>

To pass from these limiting examples to the general case, observe that a singular proposition ''s''&nbsp;:&nbsp;'''B'''^''k''&nbsp;→&nbsp;'''B''' can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>.  Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is 1 on the points adjacent to the point where ''s'' is 1, and 0 everywhere else on the cube.

To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>.  Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is 1 on the points adjacent to the point where <math>s\!</math> is 1, and 0 everywhere else on the cube.

For example, consider the case where ''k'' = 3.  Then the minimal negation operation <math>\nu (p, q, r)\!</math>, when there is no risk of confusion written more simply as <math>(p, q, r)\!</math>, has the following venn diagram:

For example, consider the case where ''k'' = 3.  Then the minimal negation operation <math>\nu (p, q, r)\!</math>, when there is no risk of confusion written more simply as <math>(p, q, r)\!</math>, has the following venn diagram: