Soft Numerical Computing in Uncertain Dynamic Systems

2.5.9.8: Generalized division

Now like generalized difference, for any two arbitrary fuzzy numbers, A and B, the generalized division can be defined as follows:

$A{\oslash }_{g}B=C\iff \left\{\begin{array}{c}i)A=B\odot C\\ \text{or}\\ \mathit{ii})B=A\odot C^{-1}\end{array}\right.$

It is clear that if both cases are true then A = B ⊙ C = A ⊙ C^− 1 ⊙ C and in this case the only option is C^− 1 ⊙ C = C ⊙ C^− 1 = {1}, where 1 is a singleton fuzzy number or a real scalar, and C is also a nonzero real scalar.

The generalized division in the level-wise or interval form can be defined as follows:

$A\left[r\right]/B\left[r\right]=C\left[r\right]\iff \left\{\begin{array}{c}i)A\left[r\right]=B\left[r\right]\cdot C\left[r\right]\\ \text{or}\\ \mathit{ii})B\left[r\right]=A\left[r\right]\cdot C^{-1}\left[r\right]\end{array}\right.$

where $C^{-1}\left[r\right]$