Abstract
Bipolar soft set is formulated by two soft sets; one of them provides us the positive information and the other provides us the negative information. The philosophy of bipolarity is that human judgment is based on two sides, positive and negative, and we choose the one which is stronger. In this paper, we introduce novel belong and nonbelong relations between a bipolar soft set and an ordinary point. These relations are considered as one of the unique characteristics of bipolar soft sets which are somewhat expression of the degrees of membership and nonmembership of an element. We discuss essential properties and derive the sufficient conditions of some equivalence of these relations. We also define the concept of soft mappings between two classes of bipolar soft sets and study the behaviors of an ordinary point under these soft mappings with respect to all relations introduced herein. Then, we apply bipolar soft sets to build an optimal choice application. We give an algorithm of this application and show the method for implementing this algorithm by an illustrative example. In conclusion, it can be noted that the relations defined herein give another viewpoint to explore the concepts of bipolar soft topology, in particular, soft separation axioms and soft covers.
Highlights
Many problems in engineering, artificial intelligence, economy, environmental science, social science, etc. involve data that contain ambiguity/vagueness. erefore, traditional methods which were based on the exact case may not be convenient for solving or modeling them.From this point, the need of new theories that help to surpass these types of instabilities arose
With the passage of time, engineers and mathematicians found alternative approaches to solve the problems that contain ambiguity/ vagueness such as probability theory, fuzzy set [1], intuitionistic fuzzy set [2], and rough set [3]. All these tools require the prespecification of some parameters to start with, for example, an equivalence relation in rough set theory and density function in probability theory
According to the fuzzy set theory, the difficulties in many problems appear in two sides: the first one is how we can determine a membership function for each particular case, and the second difficulty is the extremely individual characteristic of a membership function. at is, everyone understands the meaning of the membership function equal to 0.85 in his own manner
Summary
Artificial intelligence, economy, environmental science, social science, etc. involve data that contain ambiguity/vagueness. erefore, traditional methods which were based on the exact case may not be convenient for solving or modeling them. In 2018, the authors of [32] came up with the idea of partial belong and total nonbelong relations between an ordinary point and soft set which somewhat indicate the degree of membership and nonmembership of an element These relations widely open the door to the studying and redefining of many soft topological notions and the obtaining of many fruitful properties. We define the concept of soft mappings between two classes of weak bipolar soft sets and discuss the relationship between an ordinary point and its image and preimage with respect to the different types of belong and nonbelong relations.
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