ISSN 2071-8594

Russian academy of sciences


Gennady Osipov

Vassil Sgurev An Approach to Constructing Sets with Imaginary and Complex Elements


When solving equations with sets problems arise that there is no solution for some of them in the framework of the classic Cantor’s set theory. Introduction of an imaginary set, different from the real (classical) set is proposed in the present work. At that the union of an element or a set located in a real state with the same element or set from the imaginary state result in an empty state. This means that one and the same element in a given set may be in one only state – real or imaginary but not in both simultaneously. The set in which different elements in one of the two possible states are contained is called a complex set. With regard to these features the classical operations - ?, ?, \, ?, are adapted to complex ones. Some relations are obtained characteristic to the complex sets only. It is shown that these sets observe the algebraic requirements intrinsic to the Boolean algebra and the lattice, De Morgan’s laws for double negation, commutativity, distributivity and other. An example is shown for constructing a boolean of the complex sets.


sets-real, imaginary, complex; complex sets operations; Boolean algebra; lattice.

PP. 3-8.


1. Fraenkel A., Bar-Hillel Y, Levy A. Foundations of Set Theory, -Amsterdam, North Holland, 1973.
2. Kuratowski. Introduction to Set Theory and Topology, PSP, Warszawa, 1979.
3. Aleksandrov P. Vvedenie v teorii mnozhestv i obshchuyu topologiyu, Izd. LAN, M. 2010 (in Russian).
4. Arnold V. Geometriya komplesnykh chisel, kvaternioronov i spinov, Izd. MTsNMO, M., 2012 (in Russian).
5. Kuratowski K., Mostovski A. Set Theory, North Holland, Amsterdam, PSP, Warszawa, 1967.
6. Petrovsky A. An Axiomatic Approach to Metrization of Multiset Space, Tzeng G., Wang H., Wen U., Yu P. (Eds.) Multiple Criteria Decision Making, Springer Verlag, N.Y., 1994, pp. 129-140.
7. Sgurev V. An Essay on Complex Valued Propositional Logic, Artificial Intelligence and Decision Making, #2, 2014, pp. 85-101.
8. Vopenka. Mathematics in the Alternative Set Theory, Trubner-Texth, Leipzig, 1979.