alphabet

an alphabet is a finite, non-empty set of symbols. conventionally, we use the symbol $\text{[math]}$ for an alphabet. common alphabets include:

$\text{[math]}$ , the binary alphabet.
$\text{[math]}$ , the set of all lower-case letters.
the set of all ASCII characters, or the set of all printable ASCII characters.

[cite:@john_automata_2006]

set operations apply to alphabets.

if $\text{[math]}$ is an alphabet, we can express the set of all strings of a certain a certain length from that alphabet by using an exponential notation. we define $\text{[math]}$ to be the set of strings of length $\text{[math]}$ , each of whose symbols is in $\text{[math]}$ .
the set of all strings over an alphabet $\text{[math]}$ is conventionally denoted $\text{[math]}$ . for instance, $\text{[math]}$ . put another way,
$\text{[math]}$ sometimes we wish to exclude the empty string from te set of strings. the set of nonempty strings from alphabet $\text{[math]}$ is denoted $\text{[math]}$ . thus, two appropriate equivalences are:

$\text{[math]}$ .
$\text{[math]}$ .

[cite:@john_automata_2006 chapter 1.5 the central cocepts of automata theory]

an alphabet is any finite set.
[cite:;taken from @kozen_automata_1997 definition 2.1]