Intelligence Semantics

Distances and Similarities in Intuitionistic Fuzzy Sets by Eulalia Szmidt

By Eulalia Szmidt

This booklet offers the state of the art in thought and perform concerning similarity and distance measures for intuitionistic fuzzy units. Quantifying similarity and distances is essential for lots of purposes, e.g. info mining, laptop studying, determination making, and keep an eye on. The paintings offers readers with a accomplished set of theoretical thoughts and functional instruments for either defining and selecting similarity among intuitionistic fuzzy units. It describes an automated set of rules for deriving intuitionistic fuzzy units from info, which could reduction within the research of data in huge databases. The publication additionally discusses different vital functions, e.g. using similarity measures to guage the level of contract among specialists within the context of selection making.

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A distance on a set X is a positive function d (also called metric) from pairs of elements of X to the set R+ of non-negative real numbers with the following properties, valid for all x1 , x2 , x3 ∈ X: 1. 2. 3. 4. d(x1 , x1 ) = 0 (reflexivity); d(x1 , x2 ) = 0 if and only if x1 = x2 (separability); d(x1 , x2 ) = d(x2 , x1 ) (symmetry); d(x1 , x3 ) ≤ d(x1 , x2 ) + d(x2, x3 ) (triangle inequality). The pair (X, d) is called metric space. E. 1007/978-3-319-01640-5_3, c Springer International Publishing Switzerland 2014 39 40 3 Distances If a measure fulfills requirements 1, 3 and 4, it is called a pseudometric (separability does not hold).

7. 7. 75) ∑ πA (x). 76) by an integral over X. 79) where card(X) is the cardinality of the set X. The above norms have similar properties. g. Kaufmann [99]) two different types of distances are defined, generated from the following metric mA (x, y) = μA (x) − μA (y) and the Hamming and Euclidean metrics coincide (Atanassov [15]). 8. For an intuitionistic fuzzy set A the Hamming metric is defined as (Atanassov [15]): 1 hA (x, y) = (| μA (x) − μA(y) + νA (x) − νA (y) |). 9. For an intuitionistic fuzzy set A the the Euclidean metric is defined as (Atanassov [15]): eA (x, y) = 1 ((μA (x) − μA(y))2 + (νA (x) − νA (y))2 ).

A distance on a set X is a positive function d (also called metric) from pairs of elements of X to the set R+ of non-negative real numbers with the following properties, valid for all x1 , x2 , x3 ∈ X: 1. 2. 3. 4. d(x1 , x1 ) = 0 (reflexivity); d(x1 , x2 ) = 0 if and only if x1 = x2 (separability); d(x1 , x2 ) = d(x2 , x1 ) (symmetry); d(x1 , x3 ) ≤ d(x1 , x2 ) + d(x2, x3 ) (triangle inequality). The pair (X, d) is called metric space. E. 1007/978-3-319-01640-5_3, c Springer International Publishing Switzerland 2014 39 40 3 Distances If a measure fulfills requirements 1, 3 and 4, it is called a pseudometric (separability does not hold).

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