Changes

<br>

<br>

The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as in Table&nbsp;60.  To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.

 

The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60.  To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.

<br>

<br>

<br>

<br>

Figure&nbsp;61 shows how we might paint a picture of the logical transformation <math>F\!</math> on the canvass that was earlier primed for this purpose (way back in Figure&nbsp;30).

Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.

<br>

<br>

<br>

<br>

Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math>  The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.

Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math>  The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.

<br>

<br>

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^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</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^\bullet \to X^\bullet.\!</math>  The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math>

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^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</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^\bullet \to X^\bullet.\!</math>  The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math>