Methods of Expert Information Formalization Based on Complete Orthogonal Semantic Spaces - Expert Fuzzy Information Processing - page 48

⎧ a

⎛ ⎞ m − 1

0 0 ≤ x ≤ ⎜ ⎝ 1 − a − ⎟ ⎠ ;

⎪

m

2

⎪

a ⎞

⎛

⎪

1 a

m − 1

x

⎜ − + − ⎟

⎪ a a

m

⎛ ⎞ ⎛ ⎞

2

()

⎜ ⎟

μ x = L , ⎜ ⎝ 1 − a −

m − 1

⎟ ⎠ < x ≤ ⎜ ⎝ 1 − a +

m − 1

⎟ ⎠ ;

⎨

m m m

⎜ ⎜

a

⎟ ⎟

2 2

⎪

m − 1

⎪ ⎝ ⎠

⎪

a

⎛ ⎞

1 1 − a +

m − 1

< x ≤ 1 .

⎜ ⎝ ⎟ ⎠

⎪ m

2

⎩

Let us construct membership function of term X :

m − 1

( )

a ≥ max a , a 1. If , then

m − 1 m m − 2

⎧

⎛

m − 3

a ⎞

∑

0 0 ≤ x ≤ ⎜

⎝

a +

m − 2

⎟
⎠

;

⎪

l

2

⎪

l = 1

⎪ ⎛

m − 2

a ⎞

∑

m − 2

⎜ a + − x ⎟

⎪

l

2 ⎛

m − 3

a ⎞ ⎛

m − 2

a ⎞

⎜ ⎟

∑ ∑

⎪

l = 1 m − 2 m − 2

R , ⎜

⎝

a + ⎟

⎠

< x ≤ ⎜

⎝

a + ⎟

⎠

;

⎜ ⎟

l l

a 2 2

⎪

l = 1 l = 1

m − 2

⎜ ⎟

⎪

⎝ ⎠

⎪ ⎪

⎛

m − 2

a ⎞ 3 a

()

⎛ ⎞

μ x =

∑

⎨

1 ⎜

⎝

a +

m − 2

⎟
⎠

< x ≤ 1 −

m

;

⎜ ⎝ ⎟ ⎠

m − 1

l

2 2

⎪

l = 1

⎪

⎛ 3 a ⎞ m

⎪ ⎜ x − 1 + ⎟

3 a a

⎛ ⎞ ⎛ ⎞

2

⎪ ⎜ ⎟

−

m

< ≤ −

m

L ⎜ ⎝ 1 ⎟ ⎠ x ⎜ ⎝ 1 ⎟ ⎠ ;

⎪

⎜ ⎜

a

⎟ ⎟

2 2

m

⎪

⎝ ⎠

⎪

a

⎛ ⎞

⎪ ⎪

0 1 −

m

< x ≤ 1

⎜
⎝

⎟
⎠

2

⎩

Fig. 2.2 shows membership functions of a term for the special case

X

m − 1

() () L x = 1 − x R x = 1 − x

; ;

0 ≤ x ≤ 1

.