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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:

sg(x) = { 1 if x > 0
0 if x ≤ 0
sg(x) = { 1 if x < 0
0 if x ≥ 0

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), sg(μB(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), sg(μB(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), sg(νB(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), sg(μB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E}
—17′ {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(νB(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) + sg(νB(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) + sg(μB(x))), max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(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) + sg(μB(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))) + sg(νB(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))) + sg(νB(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), sg(μB(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), sg(μB(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), sg(νB(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), sg(μB(x))) max(νA(x), sg(1 - μB(x)))
—17′ min(μA(x), sg(1 - νB(x))) max(νA(x), sg(νB(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) + sg(νB(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) + sg(μB(x))) max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(x))) + sg(1 - μB(x)))
—32′ min(μA(x), (1 - μB(x)).μB(x) max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(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))) + sg(νB(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))) + sg(νB(x)))

Approaches to defining intuitionistic fuzzy subtractions

References

See also

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