President
"Long-term research should be conducted with a view to opening up new scientific fields and introducing the latest developments in science and technology widely in the national economy."
With the development of science and technology and further research into individual scientific fields, the mathematical methods are widely applied not only in natural sciences but also in various fields of social science including linguistics.
The purpose of introduction of mathematical methods is basically to change formulated linguistic problems into one or several simpler, logistic and algorithmically soluble mathematical ones.
Mathematical linguistics, one of technical sciences, establishes the system and makes further achievements using mathematics, especially the set theory.
First of all, let us consider the definition of ensemble in view of mathematical linguistics.
Ensemble, one of the main conceptions in modern mathematics, is a primary conception which can hardly be defined by any simpler conceptions.
In general, an aggregate of certain objects is called an ensemble, and each object in the ensemble is called element of the ensemble. Likewise, in the language, the alphabet can be seen as ensemble of a certain number of letters, the word as ensemble of a limited sequence of well-ordered letters, and the sentence as a feasible combination of inflected words.
When the association of Korean alphabets is A, A={ㄱ, ㄴ, ㄷ, ㄹ, …, ㅃ, ㅆ, ㅉ, ㅏ, ㅑ, ㅓ, ㅕ, …, ㅐ, ㅒ, ㅔ, …}, i.e., A is an ensemble of 40 letters.
When B is an association of Korean consonants, B is an ensemble of 19 consonants.
When C is an association of Korean vowels, C is an ensemble of 21 vowels.
Ensembles are represented by Latin capital letters like A, B and C, when objects are denoted by small letters like a, b and c.
When there are two ensembles A and B and all the elements of A belong to ensemble B, ensemble A is called a subset of ensemble B, and is denoted as A⊆B.
In case of ensemble A of the Korean alphabets and ensemble B of the Korean consonants, formula B⊂A is established. In case of A⊆B and B⊆A, the two ensembles A and B are said to be the same and denoted as A=B. And in case of A⊆B and A≠B, A is called a true subset and is denoted as A⊂B.
Ensemble B of the Korean consonants is a true subset of ensemble A of the Korean alphabet. For example, when A is an ensemble of the students in the linguistic department and B is an ensemble of honour students in the faculty of Korean language and literature, the formulas above give us the following meanings.
A⊂B: All the students in the linguistic department are honour students.
A=B: All the honour students in the faculty of Korean language and literature study in the linguistic department and they all are honour students.
B⊂A: There are some honour students among those who study in the linguistic department.
A∩B=Ф: There aren't any honour students in the linguistic department.
Next, let us consider the denotation of the linguistic units in view of set theory.
General alphabet is a set of finite elements like letters.
For example, alphabets of Korean, English, Japanese and {0, 1, 2… 9} are alphabet.
Word is a finite sequence in alphabetical order among the general alphabet.
Numbers from 1 to n can be attached to a word.
α=α1, α2, …, αn
β=β1, β2, …, βm
The number of consonants and vowels that make a word -α is called the length of.
∥α∥=n, ∥β∥=m
The total number of words whose length is k among the general alphabet A which consists of N letters, is equal to nk.
Sentence is a finite sequence in which words of general alphabet are written in order in compliance with a certain rule. In other words, sentence can be seen as a set of the finite number of words.
A certain ensemble of words in the general alphabet A is called the language of alphabet A. In other words, language can be seen as a true subset of which comes from the alphabet ensemble A.
