Producing fuzzy inclusion and entropy measures and their application on global image thresholding

A Method of Generating Fuzzy Implications with Specific Properties

“…Thus, for all x, y ∈ [0, 1] it is Proposition 8. Let I (1) and I (2) be two fuzzy implications that satisfy (14). Then the fuzzy implication (i) I T,I (1) ,I (2) satisfies (14) if T = T M = min{x, y} and (ii) I S,I (1) ,I (2) satisfies (14) if S = S M = max{x, y}.…”

“…Moreover, according to Corollary 9,(14) is invariant only if we use an idempotent t-norm or t-conorm. On the other hand, as we mentioned before for any fuzzy implication I (1) it is I (1) = I T M (1) = I S M (1) .…”

“…On the other hand, as we mentioned before for any fuzzy implication I (1) it is I (1) = I T M (1) = I S M (1) . So another characteristic of these three sets is that if I (1) satisfies (14), then (1) }, when they are not empty, they are sets that contain only fuzzy implications that violate (14).…”

“…Consider the Łukasiewicz's implication I LK (x, y) = min{1, 1 − x + y} that satisfies (14), (16), (15), and (17) (see [1] Table 1.4). If we are looking for a weaker fuzzy implication that satisfies (15), (17) and violates (14), this could be I T LK LK , where T LK (x, y) = max{x + y − 1, 0} (see [1] If we consider T D (x, y) = 0, if x, y ∈ [0, 1) min{x, y}, otherwise (see [1]…”

“…where I RS is the Rescher's fuzzy implication (see [1] (14). Moreover, it does not satisfy (17) since I 1 (0.9, 0.8) = 1, and it is easy to prove that satisfies (16).…”