Learning Systems: From Theory to Practice pp 47-59 | Cite as

# Real, Imaginary and Complex Sets

## Abstract

Introduction of imaginary set is proposed in the present work that is different from the classical Cantor’s set which is called real in the present work. This provides a possibility at union of an element or a set which is in real state with the same element or set in imaginary state an empty set to be obtained. The requirement pointed out allows problems to be solved which have no solution in the framework of the classical Cantor’s set theory. The introduction of additional imaginary set means that one and the same element of a given set may in one only state—real or imaginary but not in both simultaneously. A set in which different elements are contained in one of the two possible states is called complex set. Analogically to the real sets operations union, intersection, negation, difference and symmetric difference in the complex sets are proposed. Sets that do not contain an element in both states simultaneously or two identical elements in different states will be called complex normed sets. Necessary and sufficient conditions are obtained for the existence of normed complex sets and relations connected with them. It is shown that the complex sets observe the algebraic requirements characteristic of Boolean algebra and lattices—De Morgan’s laws, for the double negation, commutativity, distributivity, idempotence, as well as a series of results emerging from the complements to them being defined. Analogically to the Cantor’s sets the concepts of function, binary relation and Cartesian multiplication of complex sets are defined. At multiplication the pairs of elements received are in one of the two possible states. A table of multiplication is proposed in which the state of the pairs received is defined analogically to the way in real numbers and the operation *equivalence* in propositional logic. An example is given for constructing of a boolean from a complex set. It is pointed out that these sets may be considered as a kind of extension of the classical sets.

## Keywords

Sets Real set Imaginary set Complex sets Operations on complex sets Boolean Function Binary relation## References

- 1.Sgurev, V.: An essay on complex valued propositional logic. Artif. Intel. Decision Making
**2**, 85–101. RAS, ISSN 2071-8594 (2014)Google Scholar - 2.Fraenkel, A., Bar-Hillel, Y., Levi, A.: Foundations of Set Theory, Amsterdam, North Holland (1973)Google Scholar
- 3.Kuratowski, K.: Introduction to Set Theory and Topology. Warszawa, PSP (1979)Google Scholar
- 4.Kuratowski, K., Mostowski, A.: Set Theory. North Holland, Amsterdam, PSP, Warszawa (1967)Google Scholar
- 5.Sgurev, V.: An approach for building of imaginary and complex sets. Artif. Intel. Decision Making,
**4**(2016) (in print)Google Scholar - 6.Sgurev, V.: Features of the imaginary and complex sets. Comptes Rendus de l’Aademy Bulgare des Sciences,
**1**. ISSN 1310-1331 (2017) (in print)Google Scholar - 7.Petrovsky, A.: An axiomatic approach to metrization of multiset space. In: Tzeng G., Wang U., Yu P. (eds.) Multiple Criteria Decision Making, pp. 129–140. Springer, N.Y. (1994)Google Scholar
- 8.Vopenka, P.: Mathematics in the Alternative Set Theory, Teubner-Texth, Leipzig (1979)Google Scholar