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This paper aims to propose the notion of Type-1 single-valued neutrosophic complementary  β -neighborhood (briefly, Type-1 SVN complementary  β -neighborhood) and use it to introduce a novel class of 1-single-valued neutrosophic  β -covering rough set (briefly, 1-SVN  β -CRS). Then, we will merge the 1-SVN  β -neighborhood and 1-SVN complementary  β -neighborhood to create new two models of 1-SVN  β -CRS. Furthermore, we will discuss the relationships between the present work and Wang and Zhang’s work. For further study on Type-2 Wang and Zhang’s models, we will define the 2-SVN complementary  β -neighborhood and use it to present a novel class of 2-SVN  β -CRS. Also, we combine the 2-SVN  β -neighborhood and 2-SVN complementary  β -neighborhood to investigate the new two models of 2-SVN  β -CRS. Lately, we will demonstrate two illustrative examples as real problems to show the differences between two of our approaches and Wang and Zhang’s approach.

On Different Types of Single-Valued Neutrosophic Covering Rough Set with Application in Decision-Making