12,122
edits
Changes
Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata (view source)
Revision as of 13:34, 5 March 2009
, 13:34, 5 March 2009→Note 10: markup
Given the alphabets <math>\mathcal{U} = \{ u, v \}</math> and <math>\mathcal{X} = \{ x, y \},</math> along with the corresponding universes of discourse <math>U^\circ</math> and <math>X^\circ = [\mathbb{B}^2],</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\circ \to X^\circ</math> are there?
Given the alphabets <math>\mathcal{U} = \{ u, v \}</math> and <math>\mathcal{X} = \{ x, y \},</math> along with the corresponding universes of discourse <math>U^\circ</math> and <math>X^\circ = [\mathbb{B}^2],</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\circ \to X^\circ</math> are there?
Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2,\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\circ \to X^\circ.</math>
The set of all functions of a given type is customarily denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\circ \to X^\circ) = \{ G : U^\circ \to X^\circ \},</math> and so the cardinality of this ''function space'' can most conveniently be summed up by writing:
{| align="center" cellpadding="8" width="90%"
| <math>|(U^\circ \to X^\circ)| ~=~ |(\mathbb{B}^2 \to \mathbb{B}^2)| ~=~ 4^4 ~=~ 256.</math>
|}
<pre>
<pre>
Given any transformation of this type, G : U% -> X%, the (first order)
Given any transformation of this type, G : U% -> X%, the (first order)
differential analysis of G is based on the definition of a couple of
differential analysis of G is based on the definition of a couple of