$5\mathrm{.}5$: Languages With One String

Let One$($R$),$ for a regular expression R$,$ be the predicate stating that the language L$($R$)$ has exactly one string. Let Empty$($R$)$ be the predicate stating that the language L$($R$)$ is empty.

$$

If R and S are any two regular expressions, then the following is true:

$$

One$($R$+$S$)(($One$($R$)$ and Empty$($S$))$ or $($Empty$($R$)$ and One$($S$)))$

f$()=$ a

f$($a$)=$ b

f$($b$)=$ b

f$($R$+$S$)=$ f$($R$)+$ f$($S$)$

f$($RS$)=$ f$($R$)$f$($S$)$

f$($R$*)=$ f$($R$)*$

Then for any regular expressions T and U$,$ if T and U denote the same language, then f$($T$)$ and f$($U$)$ denote the same language.

A regular expression denotes the empty language if and only if is either is$$ or is the product of$$ and another regular expression.