Line 494: |
Line 494: |
| | | |
| 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> |
− | Since G_1 and G_2 can be any propositions of the type B^2 -> B,
| |
− | there are 2^4 = 16 choices for each of the maps G_1 and G_2, and
| |
− | thus there are 2^4 * 2^4 = 2^8 = 256 different mappings altogether
| |
− | of the form G : U% -> X%. The set of all functions of a given type
| |
− | is customarily denoted by placing its type indicator in parentheses,
| |
− | in the present instance writing (U% -> X%) = {G : U% -> X%}, and so
| |
− | the cardinality of this "function space" can be most conveniently
| |
− | summed up by writing |(U% -> X%)| = |(B^2 -> B^2)| = 4^4 = 256.
| |
− |
| |
| 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 |