universal approximation of interval-valued fuzzy systems based on interval-valued implications

it is firstly proved that the multi-input-single-output (miso) fuzzy systems based on interval-valued $r$- and $s$-implications can approximate any continuous function defined on a compact set to arbitrary accuracy.  a formula to compute the lower upper bounds on the number  of interval-valued fuzzy sets needed to achieve a pre-specified approximation  accuracy for an arbitrary multivariate continuousfunction is then presented. in addition, a method to design the interval-valued fuzzy systems based on $r$- and $s$-implications in order to approximate a given continuousfunction with a required approximation accuracy is represented. finally, two numerical examples are provided to illustrate the proposed procedure.