Alphabet

An alphabet is a finite nonempty set whose elements are called symbols.

Formal definition

An alphabet is a finite nonempty set $\Sigma$. Its elements are called symbols or letters. The cardinality $|\Sigma|$ may be any positive integer; common choices include the binary alphabet $\Sigma = \{0, 1\}$ and the full ASCII set.

An alphabet carries no structure beyond set membership: symbols are distinguishable from one another but otherwise opaque. There is no order, no algebraic operation, no notion of distance between symbols. Anything built on an alphabet — strings, languages, automata — imposes its own structure on top.

Intuition

Alphabet is the minimal formal scaffold that lets the word “symbol” mean something precise. The finiteness matters: a Turing machine’s transition table must be specifiable by a finite description, and that requires a finite domain to range over. The nonemptiness matters for less subtle reasons — over the empty alphabet the only string is the empty string, and formal language theory collapses into triviality.

The technical notion of alphabet has nothing to do with Latin alphabets or human writing systems. The binary alphabet is the one most used in practice, and single-symbol alphabets, though degenerate, are admissible.

Notes

Two concepts are built on top of this one. A string over $\Sigma$ is a finite sequence of symbols from $\Sigma$. A language over $\Sigma$ is any subset of the set of all strings over $\Sigma$. Alphabet, string, and language form the base vocabulary of automata theory and computability.