CONTRIBUTION OF PRESENT RESEARCH TO UNCERTAINTY ASSESSMENT METHODS - Modelling Uncertainty in Flood Forecasting Systems - page 71

Figure 4.14. Summation of two fuzzy variables with constant MFs (derived by
the Extension Principle).

In order to illustrate this, consider two random variables X and Y and two fuzzy variables
and ( . Let Z and be random and fuzzy variables, respectively, such that

Z = X + Y

(4.40)

(4.41)

Note that in Equation (4.41) for the addition between two fuzzy variables, the addition
symbol is enclosed by parentheses to distinguish it from the addition between random
variables. Four basic operations on fuzzy variables are presented in Appendix I. For any
value of ! -cuts ( ! =[0, 1]), the addition of two fuzzy variables (Fig. 4.15), that is

in

Equation (4.41) is given by

(4.42)

Equation (4.42) can also be written as following