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Alphabet

An alphabet is a finite set of symbols, including the empty set symbols forming the empty alphabet.  Finite sequences juxtaposed  of symbols of an alphabet are called words or strings .

The above definition of  an alphabet is an abstraction of standardized set of letters, which are used to represent phonemes of a spoken, extinct or a dead language . There are also systems of writing, in which each character represents each character represents a syllable or a word.

Since alphabet is a set we can apply set operations to form new alphabets. For instance,  if and are two alphabets, then we can define their union , intersection , difference , etc.    

We can also compare two alphabets using set inclusion as .

Often it is useful to order the symbols of an alphabet as we know it from ordering of the letters. A collation system for multiple-character words can be then induced from the ordering of symbols in the underlying alphabet. The principle behind extension of the alphabetical order to words, called the lexicographical order (or dictionary order) ¹ , is that the words are ordered  in a such a way that each th letter of the compared words are compared with the th letter of other words starting at the first letter of each word and advancing to the second, third, fourth, and so on, until we arrive at the point where the letters differ. Then the words beginning with the maximal  group of  common letters are grouped together and the ordering principle between the words is applied at the point where the letters differ.

The word alphabet  originate in the late Latin alphabetum, which in turn has origin in the Ancient Greek alphabetos, from alpha and beta, the first two letters of the Greek alphabet.

Notes

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