Enhancing rheumatic fever analysis via tritopological approximation spaces for data reduction. (English) Zbl 07894871
Summary: This paper introduces the concept of tritopological approximation space, extending conventional approximation space by drawing upon topological spaces and precisely defined binary relations within a universe of discourse. Through meticulous construction of subbases, this progressive paradigm shift facilitates a comprehensive analysis of rough sets within the domain of tritopological approximation spaces. Additionally, the study pioneer’s multiple membership functions and inclusion functions, enhancing the analytical framework and enabling more effective redefinition of rough approximations. To illustrate the practical advantages, real-life application examples are presented, focusing on the implementation of data reduction methods within the context of rheumatic fever – a prevalent disease characterized by diverse symptoms among patients, despite a consistent diagnosis. This research contributes to the advancement of rough set theory and its applications in addressing complex, real-world problems.
References:
| [1] | If T ri T (A) = T ri T (A) (i.e. T ri T BON (A) = ∅), then A is called tri-definable set. |
| [2] | If T ri T j (A) = T ri T j (A) (i.e. T ri T BON j (A) = ∅), then A is called tri-j-definable set. |
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| [5] | A set A is labeled roughly tri-j-definable if both T ri T j (A) ̸ = ∅ and T ri T j (A) ̸ = U . |
| [6] | A set A is termed internally tri-undefinable if both T ri T (A) = ∅ and T ri T (A) ̸ = U . |
| [7] | A set A is referred to as internally tri-j-undefinable if both T ri T j (A) = ∅ and T ri T j (A) ̸ = U . |
| [8] | A set A is termed externally tri-undefinable if both T ri T (A) ̸ = ∅ and T ri T (A) = U . |
| [9] | A set A is referred to as externally tri-j-undefinable if both T ri T j (A) ̸ = ∅ and T ri T j (A) = U . |
| [10] | A set A is termed totally tri-undefinable if both T ri T (A) = ∅ and T ri T (A) = U . |
| [11] | A set A is referred to as totally tri-j-undefinable if both T ri T j (A) = ∅ and T ri T j (A) = U . Remark 4.1. For any tritopological approximation space (U, R, τ r , τ ℓ , τ rℓ ). The following hold: 1. Tri -δβRD(U ) ⊇ Tri -βRD(U ) ⊇ Tri -RD(U ). |
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| [13] | Tri -δβEU D(U ) ⊆ Tri -βEU D(U ) ⊆ Tri -EU D(U ). |
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