Changes

The ''kleene plus'' <math>\mathfrak{A}^+</math> of an alphabet <math>\mathfrak{A}</math> is the set of all positive length strings over <math>\mathfrak{A},</math> in other words, everything in <math>\mathfrak{A}^*</math> but the empty string.

The ''kleene plus'' <math>\mathfrak{A}^+</math> of an alphabet <math>\mathfrak{A}</math> is the set of all positive length strings over <math>\mathfrak{A},</math> in other words, everything in <math>\mathfrak{A}^*</math> but the empty string.

A ''formal language'' <math>\mathcal{L}</math> over an alphabet <math>\mathfrak{A}</math> is a subset of <math>\mathfrak{A}^*.</math>  In brief, <math>\mathcal{L} \subseteq \mathfrak{A}^*.</math>  If <math>s\!</math> is a string over <math>\mathfrak{A}</math> and if <math>s\!</math> is an element of <math>\mathcal{L},</math> then it is customary to call <math>s\!</math> a ''sentence'' of <math>\mathcal{L}.</math>  Thus, a formal language <math>\mathcal{L}</math> is defined by specifying its elements, which amounts to saying what it means to be a sentence of <math>\mathcal{L}.</math>

A ''formal language'' <math>\mathfrak{L}</math> over an alphabet <math>\mathfrak{A}</math> is a subset of <math>\mathfrak{A}^*.</math>  In brief, <math>\mathfrak{L} \subseteq \mathfrak{A}^*.</math>  If <math>s\!</math> is a string over <math>\mathfrak{A}</math> and if <math>s\!</math> is an element of <math>\mathfrak{L},</math> then it is customary to call <math>s\!</math> a ''sentence'' of <math>\mathfrak{L}.</math>  Thus, a formal language <math>\mathfrak{L}</math> is defined by specifying its elements, which amounts to saying what it means to be a sentence of <math>\mathfrak{L}.</math>

One last device turns out to be useful in this connection.  If <math>s\!</math> is a string that ends with a sign <math>t,\!</math> then <math>s \cdot t^{-1}</math> is the string that results by ''deleting'' from <math>s\!</math> the terminal <math>t.\!</math>

One last device turns out to be useful in this connection.  If <math>s\!</math> is a string that ends with a sign <math>t,\!</math> then <math>s \cdot t^{-1}</math> is the string that results by ''deleting'' from <math>s\!</math> the terminal <math>t.\!</math>