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Fuzzy approximation based on τK fuzzy open (closed) sets

  • *Corresponding author: Priti

    *Corresponding author: Priti 
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  • Rough set theory can be generalized by induced topology through equivalence relations. Motivated by the work of generalization of the rough set via topology, the concept and properties of τK-fuzzy open (closed) sets are proposed. Considering the τK-fuzzy open (closed) sets, we have obtained the τK-fuzzy lower and upper approximations and also proved their properties. τK-fuzzy open sets can be represented as τK-open sets by α level sets. The properties of τK-fuzzy approximations and fuzzy rough approximations on the basis of binary fuzzy relation are compared. Finally, an example and the decision method's algorithm to illustrate the τK-fuzzy approximation-based approach to decision making are presented.

    Mathematics Subject Classification: Primary: 54A05, 54A40; Secondary: 03E72.

    Citation:

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  • Table 1.   

    A=P(Z1) cl(A) τ1K-FOS or not τ1K-FCS K_K(A) ¯KK(A)
    {.4u1+0u2+0u3} {.5u1+.6u2+0u3} No - {0u1+0u2+0u3} {1u1+1u2+1u3}
    {0u1+.6u2+0u3} {.3u1+.6u2+0u3} No - {0u1+0u2+0u3} {1u1+1u2+1u3}
    {0u1+0u2+.3u3} {0u1+0u2+.3u3} No - {0u1+0u2+0u3} {1u1+1u2+1u3}
    {.4u1+.6u2+0u3} {.5u1+.6u2+0u3} No - {0u1+0u2+0u3} {1u1+1u2+1u3}
    {0u1+.6u2+.3u3} {.3u1+.6u2+.3u3} No - {0u1+0u2+0u3} {1u1+1u2+1u3}
    {.4u1+0u2+.3u3} {.5u1+.6u2+.3u3} No - {0u1+0u2+0u3} {1u1+1u2+1u3}
    {.4u1+.6u2+.3u3} {.5u1+1u2+.3u3} No - {0u1+0u2+0u3} {1u1+1u2+1u3}
     | Show Table
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    Table 2.   

    A=P(Z1) cl(A) τ2K-FOS or not τ2K-FCS K_K(A) ¯KK(A)
    {.4u1+0u2+0u3} {.5u1+.6u2+.3u3} Yes {.6u1+1u2+1u3} {.4u1+0u2+0u3} {.6u1+.4u2+.7u3}
    {0u1+.6u2+0u3} {.5u1+.6u2+.3u3} Yes {1u1+.4u2+1u3} {0u1+.6u2+0u3} {.6u1+1u2+.7u3}
    {0u1+0u2+.3u3} {0u1+0u2+.3u3} No - {0u1+0u2+0u3} {.6u1+.4u2+.7u3}
    {.4u1+.6u2+0u3} {.5u1+.6u2+.3u3} Yes {.6u1+.4u2+1u3} {.4u1+.6u2+0u3} {.6u1+1u2+.7u3}
    {0u1+.6u2+.3u3} {.5u1+.6u2+.3u3} Yes {1u1+.4u2+.7u3} {0u1+.6u2+.3u3} {.6u1+1u2+.7u3}
    {.4u1+0u2+.3u3} {.5u1+.6u2+.3u3} Yes {.6u1+1u2+.7u3} {.4u1+0u2+.3u3} {.6u1+.4u2+.7u3}
    {.4u1+.6u2+.3u3} {.5u1+.6u2+.3u3} Yes {.6u1+.4u2+.7u3} {.4u1+.6u2+.3u3} {.6u1+1u2+.7u3}
     | Show Table
    DownLoad: CSV

    Table 3.   

    A=P(Z1) cl(A) τK-FOS or not τK-FCS K_K(A) ¯KK(A)
    {.4u1+0u2+0u3} {.5u1+.6u2+0u3} Yes {.6u1+1u2+1u3} {.4u1+0u2+0u3} {.6u1+.4u2+.7u3}
    {0u1+.6u2+0u3} {.3u1+.6u2+0u3} No - {0u1+0u2+0u3} {.6u1+1u2+.7u3}
    {0u1+0u2+.3u3} {0u1+0u2+.3u3} No - {0u1+0u2+0u3} {.6u1+.4u2+.7u3}
    {.4u1+.6u2+0u3} {.5u1+.6u2+0u3} Yes {.6u1+.4u2+1u3} {.4u1+.6u2+0u3} {.6u1+1u2+.7u3}
    {0u1+.6u2+.3u3} {.3u1+.6u2+.3u3} No - {0u1+0u2+0u3} {.6u1+1u2+.7u3}
    {.4u1+0u2+.3u3} {.5u1+.6u2+.3u3} Yes {.6u1+1u2+.7u3} {.4u1+0u2+.3u3} {.6u1+.4u2+.7u3}
    {.4u1+.6u2+.3u3} {.5u1+.6u2+.3u3} Yes {.6u1+.4u2+.7u3} {.4u1+.6u2+.3u3} {.6u1+1u2+.7u3}
     | Show Table
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    Table 4.   

    Sr. No. τK-fuzzy approximation
    Let (ˆU,τ,K) be FTAS and K be BFR on ˆU and Z1 and Z2 are two fuzzy subsets of ˆU.
    Fuzzy rough approximation[4,14]
    Let ˆU be a universal set, K be BFR on ˆU and Z1 and Z2 are two fuzzy subsets of ˆU.
    Z1Z2K_K(Z1)K_K(Z2) and ¯KK(Z1)¯KK(Z2) Z1Z2K_K(Z1)K_K(Z2) and ¯KK(Z1)¯KK(Z2)
    K_K(Z1)=(¯KK(Zc1))c K_K(Z1)=(¯KK(Zc1))c
    ¯KK(Z1)=(K_K(Zc1))c ¯KK(Z1)=(K_K(Zc1))c
    K_K(Z1Z2)=K_K(Z1)K_K(Z2) K_K(Z1Z2)=K_K(Z1)K_K(Z2)
    ¯KK(Z1Z2)=¯KK(Z1)¯KK(Z2) ¯KK(Z1Z2)=¯KK(Z1)¯KK(Z2)
    K_K(Z1)K_K(Z2)K_K(Z1Z2) K_K(Z1)K_K(Z2)K_K(Z1Z2)
    ¯KK(Z1Z2)¯KK(Z1)¯KK(Z2) ¯KK(Z1Z2)¯KK(Z1)¯KK(Z2)
    K_K(ϕ)=ϕ=¯KK(ϕ) ¯KK(ϕ)=ϕ
    K_K(ˆU)=ˆU=¯KK(ˆU) K_K(ˆU)=ˆU
    K_K(Z1)Z1¯KK(Z1) K_K(Z1)Z1¯KK(Z1), if K is reflexive.
    xi K_K(K_K(Z1))=K_K(Z1) K_K(K_K(Z1))=K_K(Z1), if K is reflexive and transitive.
     | Show Table
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    Table 5.   

    A=P(Z1) τK-FOS or not τK-FCS K_K(Ai) ¯KK(Ai) βi=K_K(Ai)+¯KK(Ai)2
    {.5j2+0j3+0j4+0j5} Yes {.5j2+1j3+1j4+1j5} {.5j2+0j3+0j4+0j5} {.5j2+.5j3+.3j4+.5j5} β1={.5j2+.25j3+.15j4+.25j5}
    {0j2+.5j3+0j4+0j5} Yes {1j2+.5j3+1j4+1j5} {0j2+.5j3+0j4+0j5} {.5j2+.5j3+.3j4+.5j5} β2={.25j2+.5j3+.15j4+.25j5}
    {0j2+0j3+.7j4+0j5} Yes {1j2+1j3+.3j4+1j5} {0j2+0j3+.7j4+0j5} {.5j2+.5j3+1j4+.5j5} β3={.25j2+.25j3+.85j4+.25j5}
    {0j2+0j3+0j4+.5j5} Yes {1j2+1j3+1j4+.5j5} {0j2+0j3+0j4+.5j5} {.5j2+.5j3+.3j4+.5j5} β4={.25j2+.25j3+.15j4+.5j5}
    {.5j2+.5j3+0j4+0j5} Yes {.5j2+.5j3+1j4+1j5} - - -
    {.5j2+0j3+.7j4+0j5} Yes {.5j2+1j3+.3j4+1j5} - - -
    {.5j2+0j3+0j4+.5j5} Yes {.5j2+1j3+1j4+.5j5} - - -
    {0j2+.5j3+.7j4+0j5} Yes {1j2+.5j3+.3j4+1j5} - - -
    {0j2+.5j3+0j4+.5j5} Yes {1j2+.5j3+1j4+.5j5} - - -
    {0j2+0j3+.7j4+.5j5} Yes {1j2+1j3+.3j4+.5j5} - - -
    {.5j2+.5j3+.7j4+0j5} Yes {.5j2+.5j3+.3j4+1j5} - - -
    {.5j2+.5j3+0j4+.5j5} Yes {.5j2+.5j3+1j4+.5j5} - - -
    {.5j2+0j3+.7j4+.5j5} Yes {.5j2+1j3+.3j4+.5j5} - - -
    {0j2+.5j3+.7j4+.5j5} Yes {1j2+.5j3+.3j4+.5j5} - - -
    {.5j2+.5j3+.7j4+.5j5} Yes {.5j2+.5j3+.3j4+.5j5} - - -
     | Show Table
    DownLoad: CSV
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