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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 13:40, 27 April 2012
, 13:40, 27 April 2012→6.7. Basic Notions of Formal Language Theory: markup
| <math>\underline{\underline{X}}^k ~=~ \{ (u_1, \ldots, u_k) : u_i \in \underline{\underline{X}}, i = 1 ~\text{to}~ k \}.</math>
| <math>\underline{\underline{X}}^k ~=~ \{ (u_1, \ldots, u_k) : u_i \in \underline{\underline{X}}, i = 1 ~\text{to}~ k \}.</math>
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|}
By convention for the case where <math>k = 0,\!</math> this gives <math>\underline{\underline{X}}^0 = \{ () \},</math> that is, the singleton set consisting of the empty sequence. Depending on the setting, the empty sequence is referred to as the ''empty word'' or the ''empty sentence'', and is commonly denoted by an epsilon <math>{}^{\backprime\backprime} \varepsilon {}^{\prime\prime}</math> or a lambda <math>{}^{\backprime\backprime} \lambda {}^{\prime\prime}.</math> In this text, I use the symbol "!" (a stricken exclamation point) as a synonym for the empty sequence <math>().\!</math> In addition, I use the symbol "!" (a stricken exclamation point underscored) as a synonym for <math>\{ () \}.\!</math>
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It is probably worth remarking at this point that all empty sequences are indistinguishable (in a one level formal language, that is), and thus that all singleton sets consisting of an empty sequence are identical. Consequently, X0 = {<>} = ! = Y0, for all resources X and Y. However, the empty language {} and the singleton empty sequence {<>} need to be distinguished from each other.
It is probably worth remarking at this point that all empty sequences are indistinguishable (in a one level formal language, that is), and thus that all singleton sets consisting of an empty sequence are identical. Consequently, X0 = {<>} = ! = Y0, for all resources X and Y. However, the empty language {} and the singleton empty sequence {<>} need to be distinguished from each other.