Among the basic operations which can be performed on fuzzy sets are the operations of union, intersection, complement, algebraic product and algebraic sum. In addition to these operations, new operations called “bounded-sum” and “bounded-difference” were defined by L. A. Zadeh to investigate the fuzzy reasoning which provides a way of dealing with the reasoning problems which are too complex for precise solution. This paper investigates the algebraic properties of fuzzy sets under these nex operations of bounded-sum and bounded-difference and the properties of fuzzy sets in the case where these new operations are combined with the well-known operations of union, intersection, algebraic product and algebraic sum.