Regular languages

$$(-+\Lambda) D D^{\ast} (\Lambda+.D^{\ast}), D \ \text{stands for digit}$$

Basis: $$\emptyset, \{\Lambda\} \text{ and } \{a\} \text{ are regular languages for all } a \in \Sigma$$

The basis of the definition gives us the following for regular languages over the alphabet $\Sigma = \{a,b\}$

All regular languages $\Sigma$ can be built from combining these four in various ways by recursively using the union, product and closure operation.

• $\{\Lambda,b\}$ is regular: $\{\Lambda\} \cup \{b\} = \{\Lambda, b\}$

Any finite languages is easy to build in this way $\implies$ All finite languages are regular!