Definition:Set of Finite Strings

Definition

Let $\Sigma$ be an alphabet.

The set of all finite strings from $\Sigma$ is denoted $\Sigma^*$.

Also defined as

Some sources use $\Sigma^*$ to denote the set of all strings from $\Sigma$ whether finite or not.

Examples

Over One Element

Let $\Sigma$ be the alphabet defined as:

$\Sigma = \set a$

Then the set of finite strings $\Sigma^*$ over $\Sigma$ is:

$\Sigma^* = \set {\epsilon, a, aa, aaa, aaaa, \ldots}$

where $\epsilon$ denotes the null string.

Over Two Elements

Let $\Sigma$ be the alphabet defined as:

$\Sigma = \set {0, 1}$

Then the set of finite strings $\Sigma^*$ over $\Sigma$ is:

$\Sigma^* = \set {\epsilon, 0, 1, 00, 01, 10, 11, 000, 001, \ldots}$

where $\epsilon$ denotes the null string.

Also see

• Results about sets of finite strings can be found here.

Sources

• 1988: Dominic Welsh: Codes and Cryptography ... (previous) ... (next): Notation: Alphabets and strings