Alphabet (formal languages)

In formal language theory, an alphabet, sometimes called a vocabulary, is a non-empty set of indivisible symbols/characters/glyphs,^[1] typically thought of as representing letters, characters, digits, phonemes, or even words.^[2]^[3] Alphabets in this technical sense of a set are used in a diverse range of fields including logic, mathematics, computer science, and linguistics. An alphabet may have any cardinality ("size") and, depending on its purpose, may be finite (e.g., the alphabet of letters "a" through "z"), countable (e.g., $\{v_{1},v_{2},\ldots \}$ ), or even uncountable (e.g., $\{v_{x}:x\in \mathbb {R} \}$ ).

Strings, also known as "words" or "sentences", over an alphabet are defined as a sequence of the symbols from the alphabet set.^[4] For example, the alphabet of lowercase letters "a" through "z" can be used to form English words like "iceberg" while the alphabet of both upper and lower case letters can also be used to form proper names like "Wikipedia". A common alphabet is {0,1}, the binary alphabet, and a "00101111" is an example of a binary string. Infinite sequences of symbols may be considered as well (see Omega language).

It is often necessary for practical purposes to restrict the symbols in an alphabet so that they are unambiguous when interpreted. For instance, if the two-member alphabet is {00,0}, a string written on paper as "000" is ambiguous because it is unclear if it is a sequence of three "0" symbols, a "00" followed by a "0", or a "0" followed by a "00".

^ Fletcher, Peter; Hoyle, Hughes; Patty, C. Wayne (1991). Foundations of Discrete Mathematics. PWS-Kent. p. 114. ISBN 0-53492-373-9. An alphabet is a nonempty finite set the members of which are called symbols or characters.
^ Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994). Mathematical Logic (2nd ed.). New York: Springer. p. 11. ISBN 0-387-94258-0. By an alphabet ${\mathcal {A}}$ we mean a nonempty set of symbols.
^ Rosen, Kenneth H. "Discrete Mathematics and Its Applications, Seventh Edition" McGraw-Hill 2012. Pages 847-851. From page 849: "A vocabulary (or alphabet) V is a finite, nonempty set of elements called symbols. A word (or sentence) over V is a string of finite length of elements of V."
^ Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (PDF) (Third ed.). Springer. p. xx. ISBN 978-1-4419-1220-6. If 𝗔 is an alphabet, i.e., if the elements 𝐬 ∈ 𝗔 are symbols or at least named symbols, then the sequence (𝐬₁,...,𝐬_n)∈𝗔ⁿ is written as 𝐬₁···𝐬_n and called a string or a word over 𝗔.

[1] Fletcher, Peter; Hoyle, Hughes; Patty, C. Wayne (1991). Foundations of Discrete Mathematics. PWS-Kent. p. 114. ISBN 0-53492-373-9. An alphabet is a nonempty finite set the members of which are called symbols or characters.

[Ebbinghaus1994-2] Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994). Mathematical Logic (2nd ed.). New York: Springer. p. 11. ISBN 0-387-94258-0. By an alphabet ${\mathcal {A}}$ we mean a nonempty set of symbols.

[Rosen-3] Rosen, Kenneth H. "Discrete Mathematics and Its Applications, Seventh Edition" McGraw-Hill 2012. Pages 847-851. From page 849: "A vocabulary (or alphabet) V is a finite, nonempty set of elements called symbols. A word (or sentence) over V is a string of finite length of elements of V."

[Rautenberg-4] Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (PDF) (Third ed.). Springer. p. xx. ISBN 978-1-4419-1220-6. If 𝗔 is an alphabet, i.e., if the elements 𝐬 ∈ 𝗔 are symbols or at least named symbols, then the sequence (𝐬₁,...,𝐬_n)∈𝗔ⁿ is written as 𝐬₁···𝐬_n and called a string or a word over 𝗔.

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