Given a metric space X, we consider the family of all normal upper semicontinuous fuzzy sets on X, denoted by F(X), and a discrete dynamical system (X; f). In this paper, we study when (F(X); b f) is (strongly) sensitive, where b f is the Zadeh's extension of f and F(X) is equipped with different metrics: The uniform metric, the Skorokhod metric, the sendograph metric and the endograph metric. We prove that the sensitivity in the induced dynamical system (K(X); f) is equivalent to the sensitivity in b f: F(X)! F(X) with respect to the uniform metric, the Skorokhod metric and the sendograph metric. We also show that the following conditions are equivalent: a) (X; f) is strongly sensitive; b) (F(X); b f) is strongly sensitive, where F(X) is equipped with the uniform metric; c) (F(X); b f) is strongly sensitive, where F(X) is equipped with the Skorokhod metric; d) (F(X); b f) is strongly sensitive, where F(X) is equipped with the sendograph metric.