Relations between complex vague soft sets

• Complex vague soft sets (CVSSs for short) are a hybrid model of complex fuzzy sets and vague soft sets and the study of CVSSs was initiated by Selvachandran et al. (2016).
• In this paper, the Cartesian product and the composition of CVSSs are defined and these are used to introduce the relations between CVSSs.
• This relations are then classified into symmetric, transitive, reflexive and equivalence relations and the algebraic properties and the relationships between these relations are then studied.
• The relations between CVSSs introduced here have the ability to: (i) Describe the degree of interaction and the phase between the elements in two sets by means of an interval-based membership structure. (ii) Capture the information pertaining to the time frame of the intersection between the elements in two sets, which cannot be accomplished by using other relations such as relations between fuzzy sets, soft sets or complex fuzzy sets.
• An application of the relations between CVSSs in the area of economics is then presented through the discussion of a real-life problem involving the influence of the financial indicators of the Chinese economy on the financial indicators of the Malaysian economy.

Complex vague soft sets are essentially vague soft sets characterized by an additional parameter called the phase term which is defined over the set of complex numbers. In this study, we introduce and discuss the relations between complex vague soft sets. We present the definitions of the Cartesian product of complex vague soft sets and subsequently that of complex vague soft relations. The definition of the composition of complex vague soft sets is also provided. The notions of symmetric, transitive, reflexive and equivalence complex vague soft relations are then proposed and the algebraic properties of these concepts are verified. The relation between complex vague soft sets is then discussed in the context of a real-life problem: the relation between the financial indicators of the Chinese economy which are characterized by their degrees of influence on the financial indicators of the Malaysian economy, and the time required for the former to affect the latter. Interpretations of the results obtained from this example are then proposed by relating them to recent significant real-life events in the Chinese and Malaysian economies.

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