Some properties of the entropy and information energy for fuzzy sets

We study the properties of entropy and information energy for fuzzy sets. The relationship between entropy and information energy is also discussed. We show that the entropy of a fuzzy set is a measure of the fuzziness of the fuzzy set. Moreover, the entropy is a global measure for the fuzzy set since it is an integral in the whole universal set. For two fuzzy sets with the same shape, the fatter set is fuzzier than the thinner set. In other words, the entropy of a fuzzy set depends very much on its size. However, the information energy of a fuzzy set is a local measure, since it is an integral in a considered interval. The larger the size of the considered interval, the more we get from the interval for the information energy. In other words, the integral interval is much more important than the support of a fuzzy set for the information energy. Furthermore, the information energy and the entropy of a fuzzy set have a trade-off relationship.