Changes

The term ''exponentiation'' is more generally used in mathematics for operations that involve taking a base to a power, and is slightly preferable to ''involution'' since the latter is used for different concepts in different contexts.  Operations analogous to taking powers are widespread throughout mathematics and Peirce frequently makes use of them in a number of important applications, for example, in his theory of information.  But that's another story.

The term ''exponentiation'' is more generally used in mathematics for operations that involve taking a base to a power, and is slightly preferable to ''involution'' since the latter is used for different concepts in different contexts.  Operations analogous to taking powers are widespread throughout mathematics and Peirce frequently makes use of them in a number of important applications, for example, in his theory of information.  But that's another story.

The ''function space'' <math>Y^X,\!</math> where <math>X\!</math> and <math>Y\!</math> are sets, is the set of all functions from <math>X\!</math> as ''domain'' to <math>Y\!</math> as ''codomain'', defined as <math>Y^X = \{f : X \to Y \}.</math> The form <math>(X \to Y)</math> is also used to denote the function space <math>Y^X.\!</math> If <math>X\!</math> and <math>Y\!</math> have cardinalities <math>|X|\!</math> and <math>|Y|,\!</math> respectively, then the function space <math>X^Y\!</math> has cardinality <math>|X|^{|Y|}.\!</math>

The ''function space'' <math>Y^X,\!</math> where <math>X\!</math> and <math>Y\!</math> are sets, is the set of all functions from <math>X\!</math> to <math>Y.\!</math> An alternative notation for <math>Y^X\!</math> is <math>(X \to Y).</math> Thus we have the following equivalents:

 

| <math>\begin{matrix} Y^X & = & (X \to Y) & = & \{f : X \to Y \} \end{matrix}</math>

 

If <math>X\!</math> and <math>Y\!</math> have cardinalities <math>|X|\!</math> and <math>|Y|,\!</math> respectively, then the function space <math>X^Y\!</math> has a cardinality given by the following equation:

 

| <math>\begin{matrix}|X^Y| & = & |X|^{|Y|}\end{matrix}</math>

==References==

==References==