8-9 October 2020 • Burgas, Bulgaria

Submission: 15 May 2020 • Notification: 31 May 2020 • Final Version: 15 June 2020

Subtractions over intuitionistic fuzzy sets

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For the various definitions of subtraction of over intuitionistic fuzzy sets, the functions sg and sg have been used:

Failed to parse (syntax error): \text{sg}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x > 0 \\ 0 & \text{if } x \leq 0 \end{array},    Failed to parse (syntax error): \overline{\text{sg}}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x < 0 \\ 0 & \text{if } x \geq 0 \end{array}.

List of intuitionistic fuzzy subtractions of —i type

Alternative separated view

List of intuitionistic fuzzy subtractions of —i′ type

No. Ref. Year Subtraction
01 {<x, min(μA(x), νB(x)), max(νA(x), μB(x))>|x ∈ E}
02 {<x, min(μA(x), sgB(x))), max(νA(x), sg(μB(x)))>|x ∈ E}
03 {<x, min(μA(x), νB(x)), max(νA(x), μB(x).νB(x) + μB(x)2)>|x ∈ E}
04 {<x, min(μA(x), νB(x)), max(νA(x), 1 - νB(x))>|x ∈ E}
05 {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(1 - νB(x)))>|x ∈ E}
06 {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(μB(x)))>|x ∈ E}
07 {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), μB(x))>|x ∈ E}
08 {<x, min(μA(x), 1 - μB(x)), max(νA(x), μB(x))>|x ∈ E}
09 {<x, min(μA(x), sgB(x))), max(νA(x), μB(x))>|x ∈ E}
10 {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), 1 - νB(x))>|x ∈ E}
11 {<x, min(μA(x), sg(νB(x))), max(νA(x), sgB(x)))>|x ∈ E}
12 {<x, min(μA(x), νB(x).(μB(x) + νB(x))), max(νA(x), μB(x).(νB(x)2 + μB(x) + μB(x).νB(x)))>|x ∈ E}
13 {<x, min(μA(x), sg(1 - μB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E}
14 {<x, min(μA(x), sg(νB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E}
15 {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E}
16 {<x, min(μA(x), sgB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E}
17 {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sgB(x)))>|x ∈ E}
18 {<x, min(μA(x), νB(x), sg(μB(x))), max(νA(x), min(μB(x), sg(νB(x))))>|x ∈ E}
19 {<x, min(μA(x), νB(x), sg(μB(x))), νA(x)>|x ∈ E}
20 {<x, min(μA(x), νB(x)), νA(x)>|x ∈ E}
21 {<x, min(μA(x), 1 - μB(x), sg(μB(x))), max(νA(x), min(μB(x), sg(1 - μB(x))))>|x ∈ E}
22 {<x, min(μA(x), 1 - μB(x), sg(μB(x))), νA(x)>|x ∈ E}
23 {<x, min(μA(x), 1 - μB(x)), νA(x)>|x ∈ E}
24 {<x, min(μA(x), νB(x), sg(1 - νB(x))), max(νA(x), min(1 - νB(x), sg(νB(x))))>|x ∈ E}
25 {<x, min(μA(x), νB(x), sg(1 - νB(x))), νA(x)>|x ∈ E}
26 {<x, min(μA(x), νB(x)), max(νA(x), μB(x).νB(x) + sg(1 - μB(x)))>|x ∈ E}
27 {<x, min(μA(x), 1 - μB(x)), max(νA(x), μB(x).(1 - μB(x)) + sg(1 - μB(x)))>|x ∈ E}
28 {<x, min(μA(x), νB(x)), max(νA(x), (1 - νB(x)).νB(x) + sgB(x)))>|x ∈ E}
29 {<x, min(μA(x), max(0, μB(x).νB(x) + sg(1 - νB(x)))), max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))>|x ∈ E}
30 {<x, min(μA(x), μB(x).νB(x), max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))>|x ∈ E}
31 {<x, min(μA(x), (1 - μB(x)).μB(x) + sgB(x))), max(νA(x), μB(x).((1 - μB(x)).μB(x) + sgB(x))) + sg(1 - μB(x)))>|x ∈ E}
32 {<x, min(μA(x), (1 - μB(x)).μB(x), max(νA(x), μB(x).((1 - μB(x)).μB(x) + sgB(x))) + sg(1 - μB(x)))>|x ∈ E}
33 {<x, min(μA(x), νB(x).(1 - νB(x)) + sg(1 - νB(x))), max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sgB(x)))>|x ∈ E}
34 {<x, min(μA(x), νB(x).(1 - νB(x))), max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sgB(x)))>|x ∈ E}

Alternative separated view

No. Ref. Year Subtraction:

{<x, Subtraction MEMBERSHIP expression, Subtraction NON-MEMBERSHIP expression >|x ∈ E}

No. Ref. Year Subtraction MEMBERSHIP expression
Subtraction NON-MEMBERSHIP expression
01 min(μA(x), νB(x)) max(νA(x), μB(x))
02 min(μA(x), sgB(x))) max(νA(x), sg(μB(x)))
03 min(μA(x), νB(x)) max(νA(x), μB(x).νB(x) + μB(x)2)
04 min(μA(x), νB(x)) max(νA(x), 1 - νB(x))
05 min(μA(x), sg(1 - νB(x))) max(νA(x), sg(1 - νB(x)))
06 min(μA(x), sg(1 - νB(x))) max(νA(x), sg(μB(x)))
07 min(μA(x), sg(1 - νB(x))) max(νA(x), μB(x))
08 min(μA(x), 1 - μB(x)) max(νA(x), μB(x))
09 min(μA(x), sgB(x))) max(νA(x), μB(x))
10 min(μA(x), sg(1 - νB(x))) max(νA(x), 1 - νB(x))
11 min(μA(x), sg(νB(x))) max(νA(x), sgB(x)))
12 min(μA(x), νB(x).(μB(x) + νB(x))) max(νA(x), μB(x).(νB(x)2 + μB(x) + μB(x).νB(x)))
13 min(μA(x), sg(1 - μB(x))) max(νA(x), sg(1 - μB(x)))
14 min(μA(x), sg(νB(x))) max(νA(x), sg(1 - μB(x)))
15 min(μA(x), sg(1 - νB(x))) max(νA(x), sg(1 - μB(x)))
16 min(μA(x), sgB(x))) max(νA(x), sg(1 - μB(x)))
17 min(μA(x), sg(1 - νB(x))) max(νA(x), sgB(x)))
18 min(μA(x), νB(x), sg(μB(x))) max(νA(x), min(μB(x), sg(νB(x))))
19 min(μA(x), νB(x), sg(μB(x))) νA(x)
20 min(μA(x), νB(x)) νA(x)
21 min(μA(x), 1 - μB(x), sg(μB(x))) max(νA(x), min(μB(x), sg(1 - μB(x))))
22 min(μA(x), 1 - μB(x), sg(μB(x))) νA(x)
23 min(μA(x), 1 - μB(x)) νA(x)
24 min(μA(x), νB(x), sg(1 - νB(x))) max(νA(x), min(1 - νB(x), sg(νB(x))))
25 min(μA(x), νB(x), sg(1 - νB(x))) νA(x)
26 min(μA(x), νB(x)) max(νA(x), μB(x).νB(x) + sg(1 - μB(x)))
27 min(μA(x), 1 - μB(x)) max(νA(x), μB(x).(1 - μB(x)) + sg(1 - μB(x)))
28 min(μA(x), νB(x)) max(νA(x), (1 - νB(x)).νB(x) + sgB(x)))
29 min(μA(x), max(0, μB(x).νB(x) + sg(1 - νB(x)))) max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))
30 min(μA(x), μB(x).νB(x) max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))
31 min(μA(x), (1 - μB(x)).μB(x) + sgB(x))) max(νA(x), μB(x).((1 - μB(x)).μB(x) + sgB(x))) + sg(1 - μB(x)))
32 min(μA(x), (1 - μB(x)).μB(x) max(νA(x), μB(x).((1 - μB(x)).μB(x) + sgB(x))) + sg(1 - μB(x)))
33 min(μA(x), νB(x).(1 - νB(x)) + sg(1 - νB(x))) max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sgB(x)))
34 min(μA(x), νB(x).(1 - νB(x))) max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sgB(x)))


List of intuitionistic fuzzy subtractions of —i′′ type

Alternative separated view

Approaches to defining intuitionistic fuzzy subtractions

References

See also

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