Language Theory: Interpreting the Regular Expression $(a \mid b)^*$

The regular expression $(a \mid b)^*$ uses two core operations: union $(a \mid b)$ and Kleene star ${}^*$. Intuitively, $(a \mid b)$ means “choose either $a$ or $b$ as the next symbol,” and ${}^*$ means “repeat that choice any number of times, including zero repetitions.” Therefore, $(a \mid b)^*$ denotes the language of all finite strings over the alphabet $\{a,b\}$, including the empty string $\varepsilon$.

A precise way to see this is: $(a \mid b)^*$ is the set of strings that can be written as concatenations of zero or more symbols, where each symbol is either $a$ or $b$—hence every string over $\{a,b\}$ is included, and $\varepsilon$ is included because zero repetitions is allowed.

We can express the option mapping directly:

• (i) “Strings containing only $a$” — only a subset of the language.
• (ii) “Strings containing only $b$” — only a subset of the language.
• (iii) “All strings over $\{a,b\}$, including $\varepsilon$” — exactly the language.
• (iv) “Strings beginning with $ab$” — only a subset (and does not include many strings that don’t begin with $ab$ like $\varepsilon$, $a$, $b$, $ba$, etc.).

Below is a structured breakdown of why.

Key terms you’ll use:

• KleeneStar
• Union
• RegularLanguage
• Epsilon
• Alphabet

Note: In regular-expression semantics, $E^*$ always includes $\varepsilon$ (zero repetitions). That single fact already rules out option (iv) and confirms option (iii).