This paper will study the arithmetic operations of non-normal fuzzy sets using the
concept of gradual numbers that can be regarded as the elements of gradual sets. We
shall present the idea in which the fuzzy sets can be formulated as consisting of
gradual elements like the usual set consisting of usual elements. When the universal
set is taken to be the real number system, the gradual element is also called a gradual
number. In this case, the arithmetic operations of non-normal fuzzy sets can be defined
by using the gradual numbers. (C) 2019 Elsevier B.V. All rights reserved.