fuzzy-set-operations.md

Fuzzy Operations

Triangular Norms (T-norms)

Four t-norms functions are implemented in the t-norms.R script:

• Minimum
• Product
• Lukasiewicz
• Drastic Product

source("t-norms.R")

A = ((min_A, mean_A, max_A)) and B = ((min_B, mean_B, max_B)) are triangular fuzzy numbers represented by three points: minimum, mean and maximum values

fuzzy.number.a = c(min.value = 1, mean.value = 4, max.value = 6)
fuzzy.number.b = c(min.value = 3, mean.value = 5, max.value = 7)

The plot.tnorms function gives the plot of all T-norms

plot.tnorms(fuzzy.number.a, fuzzy.number.b)

Minimum T-norm

plot.tnorms(fuzzy.number.a, fuzzy.number.b, "min")

Product T-norm

plot.tnorms(fuzzy.number.a, fuzzy.number.b, "pro")

Lukasiewicz T-norm

plot.tnorms(fuzzy.number.a, fuzzy.number.b, "luka")

Drastic Product T-norm

plot.tnorms(fuzzy.number.a, fuzzy.number.b, "dra")

Triangular Conorms (S-norms)

Four t-norms functions are implemented in the s-norms.R script:

• Maximum
• Probabilistic sum
• Bounded sum (Lukasiewicz)
• Drastic sum

source("s-norms.R")

The plot.snorms function gives the plot of all S-norms

plot.snorms(fuzzy.number.a, fuzzy.number.b)

Maximum S-norm

plot.snorms(fuzzy.number.a, fuzzy.number.b, "max")

Probabilistic sum S-norm

plot.snorms(fuzzy.number.a, fuzzy.number.b, "pro")

Bounded sum (Lukasiewicz) S-norm

plot.snorms(fuzzy.number.a, fuzzy.number.b, "bound")

Drastic sum S-norm

plot.snorms(fuzzy.number.a, fuzzy.number.b, "dra")