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agop (version 0.2.4)

tconorm_minimum: t-conorms

Description

Various t-conorms. Each of these is a fuzzy logic generalization of the classical alternative operation.

Usage

tconorm_minimum(x, y)

tconorm_product(x, y)

tconorm_lukasiewicz(x, y)

tconorm_drastic(x, y)

tconorm_fodor(x, y)

Value

Numeric vector of the same length as x and y. The ith element of the resulting vector gives the result of calculating S(x[i], y[i]).

Arguments

x

numeric vector with elements in [0,1]

y

numeric vector of the same length as x, with elements in [0,1]

Details

A function S:[0,1]×[0,1][0,1] is a t-conorm if for all x,y,z[0,1] it holds: (a) S(x,y)=S(y,x); (b) if yz, then S(x,y)S(x,z); (c) S(x,S(y,z))=S(S(x,y),z); (d) S(x,0)=x.

The minimum t-conorm is given by SM(x,y)=max(x,y).

The product t-conorm is given by SP(x,y)=x+yxy.

The Lukasiewicz t-conorm is given by SL(x,y)=min(x+y,1).

The drastic t-conorm is given by SD(x,y)=1 iff x,y(0,1], and max(x,y) otherwise.

The Fodor t-conorm is given by SF(x,y)=1 iff x+y1, and max(x,y) otherwise.

References

Klir G.J, Yuan B., Fuzzy sets and fuzzy logic. Theory and applications, Prentice Hall PTR, New Jersey, 1995.

Gagolewski M., Data Fusion: Theory, Methods, and Applications, Institute of Computer Science, Polish Academy of Sciences, 2015, 290 pp. isbn:978-83-63159-20-7

See Also

Other fuzzy_logic: fimplication_minimal(), fnegation_yager(), tnorm_minimum()