Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 19

An interval-based Type II fuzzy set is constructed by defining upper and
lower membership values of type I fuzzy set. So, the membership function is
in an interval range, that is, fuzzy, and is written as

upper α

μ = [()] μ x

(1.17)

lower

[()] / x 1

α

μ = μ

where 0 < α ≤ 1.
The points within the membership areas that is in the interval region are
represented by type I fuzzy set. So, a more practical form of representing
Type II fuzzy set is

A

Type II =

{, (), ()| x μμ x x xX ∈ }

U L

and

μ () () (), x < μ x < μ x μ ∈ 01 ,

⎣ ⎦

U L

where upper and lower limits may be the linguistic hedges like dilation or
concentration:

05 .

μ () () x = μ x

U

2

μ () () x = μ x

L

or it can be deaccentuation or accentuation:

075 .

μ () () x = μ x

U

125 .

μ () () x = μ x

L

As the linguistic hedges form in pairs, the practical form of representing
hedges may be written as

1 / α

μ () () x = μ x

U

α

μ () () x = μ x

L

with α ∈ [1, ∞).