DOI: https://doi.org/10.15587/2312-8372.2015.41057

### Solution of the problem of set theory on the basis of algebra of predicates and predicate operations

Наталья Тимофеевна Процай

#### Abstract

The mathematical tools of algebra of predicates and predicate operations to solve the problems in the set theory are applied in the article. The basic concepts of algebra of predicates and predicate operations, their types and formulas are considered. The quantifier algebra of predicate operations, which is complete and plays an important role in the description of operations on predicates, is considered. The basic identities and laws of quantifier algebra of predicate operations are considered. With the help of quantifier algebra of predicate operations describes the concept of linear logical operator and Galois mapping, which are usable with the capture of data quantifiers in an empty domain. The basic concepts of set theory are described. The proper interpretation of the language of algebra predicates and predicate operations received operations of union, intersection, complement, difference, symmetric difference and inclusion of sets. Some problems of set algebra are solved by means of sets of algebra of predicates and predicate operations. The advantages of using these tools to solve the problems of algebra of sets are marked.

#### Keywords

algebra of predicates and predicate operations; set algebra; set operations

PDF (Русский)

#### References

Bondarenko, M. F., Dudar', Z. V., Shabanov-Kushnarenko, Yu. P., Chikina, V. A., Protsai, N. T., Cherkashin, V. V. (2005). Algebra predikatov i predikatnyh operatsii. Radioelektronika ta informatyka, 1, 80–86.

Protsai, N. T. (2008). Kvantornaia algebra predikatnyh operatsii. Bionika intelektu, 1 (68), 69-73.

Bondarenko, M. F., Shabanov-Kushnarenko, Yu. P. (2004). Ob algebre predikatov. Bionika intellekta, 1 (61), 15-26.

Protsai, N. T., Vechirskaiach, I. D. (2014). Opredelenie obraza lineinogo logicheskogo operatora i otobrazheniia Galua po pustoi oblasti v terminah kvantornoi algebry. Bionika intellekta, 2 (83), 30-34.

Bondarenko, M. F., Shabanov-Kushnarenko, Yu. P. (2006). Teoriia intellekta. H.: SMIT, 563.

Weiss, A. R. (2008). An introduction to set theory. University of Toronto, 119.

Enderton, H. B. (1979). Elements of Set Theory. Elsevier BV, 279. doi:10.1016/B978-0-08-057042-6.50006-5

Bourbaki, N. (2004). Elements of Mathematics. Theory of Sets. Springer, 414. doi:10.1007/978-3-642-59312-3

Herrlich, H. (2006). Axiom of Choice. Lecture Notes in Mathematics. Springer Science + Business Media, 194.doi:10.1007/3-540-34268-0_4

Ciesielski, K. (1997). Set Theory for the Working Mathematician. Cambridge University Press, 252. doi:10.1017/cbo9781139173131

#### GOST Style Citations

Бондаренко, М. Ф. Алгебра предикатов и предикатных операций [Текст] / М. Ф. Бондаренко, З. В. Дударь, Ю. П. Шабанов-Кушнаренко, В. А. Чикина, Н. Т. Процай, В. В. Черкашин // Радіоелектроніка та інформатика. – 2005. – № 1. – С. 80–86.

Процай, Н. Т. Кванторная алгебра предикатных операций [Текст] / Н. Т. Процай // Біоніка інтелекту. – 2008. – № 1 (68). – С. 69-73.

Бондаренко, М. Ф. Об алгебре предикатов [Текст] / М. Ф. Бондаренко, Ю. П. Шабанов-Кушнаренко // Бионика интеллекта. – 2004. – № 1 (61). – С. 15-26.

Процай, Н. Т. Определение образа линейного логического оператора и отображения Галуа по пустой области в терминах кванторной алгебры [Текст] / Н. Т. Процай, И. Д. Вечирскаяч // Бионика интеллекта. – 2014. – № 2 (83). – С. 30-34.

Бондаренко, М. Ф. Теория интеллекта [Текст] / М. Ф. Бондаренко, Ю. П. Шабанов-Кушнаренко. – Х.: СМИТ, 2006. – 563 с.

Weiss, A. R. An introduction to set theory [Text] / A. R. Weiss. – University of Toronto, 2008. – 119 p.

Enderton, H. B. Elements of Set Theory [Text]: Hardcover / H. B. Enderton. – Elsevier BV, 1979. – 279 p. doi:10.1016/B978-0-08-057042-6.50006-5

Bourbaki, N. Elements of Mathematics. Theory of Sets [Text] / N. Bourbaki. – Springer, 2004. – 414 p. doi:10.1007/978-3-642-59312-3

Herrlich, H. Axiom of Choice [Text] / H. Herrlich // Lecture Notes in Mathematics. – Springer-Verlag, 2006. –194 p.doi:10.1007/3-540-34268-0_4

Ciesielski, K. Set Theory for the Working Mathematician [Text] / K. Ciesielski. – Cambridge University Press, 1997. – 252 p. doi:10.1017/cbo9781139173131

Copyright (c) 2016 Наталья Тимофеевна Процай