α α α 1. X A X µ A : X → [0, 1] X [0, 1] A = {(a, µ A (a))|a ∈ X} µ A µ A (a) ∈ [0, 1] a A µ A [0, 1] 0 1 A = {(a, µ A (a))|a ∈ X} µ A A µ A (a) = 0 a ∈ X. A µ A (a) = 1, a ∈ X. µ ν X µ  ν µ  ν µ = ν µ  ν µ(x)  ν(x), x ∈ X,(1.0.1) µ  ν µ(x)  ν(x), x ∈ X,(1.0.2) µ = ν µ(x) = ν(x), x ∈ X.(1.0.3) 1 x : R−→[0, 1] t∈R x(t) x t 0 x : R → [0, 1], t 0 x s  t  r x(t)  min{x(s), x(r)}.(1.0.4) t 0 ∈ R x x(t 0 ) = 1 x x x(t) = 0 t < 0 E G G ⊆ E x ∈ R x x(t) =  1 t = x, 0 t = x. +, −, ·, / E × E (x + y)(t) = sup s∈R min{x(s), y(t − s)}, ∀x, y ∈ E, ∀t ∈ R,(1.0.5) (x − y)(t) = sup s∈R min{x(s), y(s − t)}, ∀x, y ∈ E, ∀t ∈ R,(1.0.6) (x · y)(t) = sup s∈R min  x(s), y  t s  , ∀x, y ∈ E, ∀t ∈ R,(1.0.7) (x/y)(t) = sup s∈R min{x(ts), y(s)}, ∀x, y ∈ E, ∀t ∈ R.(1.0.8) x ∈ R 0 1 E 0(t) =  1 t = 0, 0 t = 0. 1(t) =  1 t = 1, 0 t = 1. x ∈ R x(t) = 0(t − x) t ∈ R. x ∈ R t ∈ R 0(t − x) =  1 t − x = 0, 0 t − x = 0. =  1 x = t, 0 x = t. = x(t)  x x |x| |x|(t) =  max{x(t), x(−t))} t  0, 0 t < 0. (1.0.9) y ∈ E 0 − y ∈ E −y x α ∈ (0, 1] α α x [x] α [x] α = {t ∈ R|x(t)  α}.(1.0.10) α  a α , b α  [x] α =  a α , b α  a α ∞ b α ∞  − ∞, b α   a α , +∞  X ∅ λ α : X × X → R X × X R α ∈ (0, 1] α 1 , α 2 ∈ (0, 1] α 1 < α 2 λ α 1 (x, y)  λ α 2 (x, y) (x, y) ∈ X × X X ∅ ρ α : X × X → R X × X R α ∈ (0, 1] α 1 , α 2 ∈ (0, 1] α 1 < α 2 ρ α 1 (x, y)  ρ α 2 (x, y) (x, y) ∈ X × X X ∅ d : X × X → G X × X G L, R : [0, 1] × [0, 1] → [0, 1] x y L(0, 0) = 0 R(1, 1) = 1  d(x, y)  α =  λ α (x, y), ρ α (x, y)  ,(1.0.11) x, y ∈ X λ ρ (X, d, L, R) (i.) d(x, y) = 0 x = y, (ii.) d(x, y) = d(y, x) x, y ∈ X, (iii.) x, y, z ∈ X (1.) d(x, y)(s + t)  L  d(x, z)(s), d(z, y)(t)  s  λ 1 (x, z) t  λ 1 (z, y) s + t  λ 1 (x, y) (2.) d(x, y)(s + t)  R  d(x, z)(s), d(z, y)(t)  s  λ 1 (x, z) t  λ 1 (z, y) s + t  λ 1 (x, y) λ α , ρ α λ α α ρ α α d (X, d) L R L(a, b) = 0 a, b ∈ [0, 1] R(a, b) =  0 a = b = 0, 1 . ∆ [0, 1] × [0, 1] → [0, 1] ∆ ∆(a, 1) = a, a ∈ [0, 1]; ∆(a, b) = ∆(b, a), a, b ∈ [0, 1]; ∆(c, d)  ∆(a, b), c  a d  b a, b, c, d ∈ [0, 1]; ∆(a, ∆(b, c)) = ∆(∆(a, b), c) a, b, c ∈ [0, 1]. F : R −→ R + F inf t∈R F (t) = 0 sup t∈R F (t) = 1 D X ∅ F : X × X → D X × X D x, y ∈ X F xy = F(x, y) (X, F ) F xy F xy (t) = 1 t > 0 x = y, F xy (0) = 0 x, y ∈ X, F xy (t) = F yx (t) t ∈ R x, y ∈ X, F xz (t) = 1 F zy (s) = 1 F xy (s + t) = 1 x, y, z ∈ X X ∅ (X, F, ∆) (X, F ) ∆ : [0, 1] × [0, 1] −→ [0, 1] t (x, y) ∈ X × X F xy F xy (0) = 0 (i)  t > 0 F xy (t) = 1 x = y, (ii)  F xy = F yx , x, y ∈ X, (iii)  F xy (s + r)  ∆  F xz (s), F zy (r)  , x, y, z ∈ X, s, r  0 L R ∆ T 1 (a, b) = max(a + b − 1, 0) (max( − 1, 0)) T 2 (a, b) = ab ( ) T 3 (a, b) = min(a, b) (min) T 4 (a, b) = max(a, b) (max) T 5 (a, b) = a + b − ab ( ) T 6 (a, b) = min(a + b, 1) (min( , 1)) T i , i = 1, 2, , 6 i  j T i (a, b)  T j (a, b) a, b ∈ [0, 1] 2. α α R x ∈ R, x(t) = 1 t = x x(t) = 0 t = x x x s  t  r, s, r ∈ R. x = t x(t) = 1 x(t)  min{x(s), x(r)} x = t, x(t) = 0 x = s x = r x = r x = s x = s x = r x(t)  min{x(s), x(r)} = 0 x R  x, y ∈ E (−y)(t) = y(−t) t ∈ R x − y = x + (−y) x α ∈ (0, 1] α [x] α R x, y ∈ E [x] α  a α 1 , b α 1  [y] α  a α 2 , b α 2  [x + y] α =  a α 1 + a α 2 , b α 1 + b α 2  ,(2.0.12) [x.y] α =  a α 1 .a α 2 , b α 1 .b α 2  , x, y ∈ G,(2.0.13) [x − y] α =  a α 1 − b α 2 , b α 1 − a α 2  ,(2.0.14) [1 x] α =  1 b α 1 , 1 a α 1  a α 1 > 0,(2.0.15)  |x|  α =  max{0, a α 1 , −b α 1 }, max{|a α 1 |, |b α 1 |}  .(2.0.16) (X, d, L, R) (1) t ∈ R + α ∈ (0, 1] d(x, y)(t)  α λ α (x, y)  t  ρ α (x, y). (2) t ∈ R + α ∈ (0, 1] d(x, y )(t) < α λ α (x, y) > t ρ α (x, y) < t. (1) t ∈ R + α ∈ (0, 1] d(x, y)(t)  α t ∈ [d(x, y)] α λ α (x, y)  t  ρ α (x, y). (2) t ∈ R + α ∈ (0, 1] d(x, y)(t) < α. t ∈ [d(x, y)] α t ∈ [λ α (x, y), ρ α (x, y)] λ α (x, y) > t ρ α (x, y) < t.  d(x, y) ∈ G x, y ∈ X G 0  d(x, y)(t)  1 x, y ∈ X t  0 3. (X, d, L, R) L R T i , i = 1, 2, , 6 F : X × X → G F(x, y)(t) =  0 1 − d(x, y)(t) = (X, F) F F xy (t) = 1, t > 0 d(x, y)(t) = 0 t > 0 d(x, y) = 0 x = y. F xy (t) = 1 t > 0 x = y F xy (0) = 0, x, y ∈ X. x, y ∈ X t ∈ R F xy (t) = 1 − d(x, y)(t) = 1 − d(y, x)(t) = F yx (t). F xy (t) = F yx (t), t ∈ R x, y ∈ X. t, s ∈ R, x, y, z ∈ X F xz (t) = 1 F zy (s) = 1 d(x, z)(t) = 0 d(z, y)(s) = 0. d(x, y)(t + s)  L  d(x, z)(t), d(z, y)(t)  = L(0, 0) = 0, t  λ 1 (x, z), s  λ 1 (z, y) t + s  λ 1 (x, y). d(x, y)(t + s)  R  d(x, z)(t), d(z, y)(t)  = R(0, 0) = 0, t  λ 1 (x, z), s  λ 1 (z, y) t + s  λ 1 (x, y). d(x, y) ∈ G d(x, y)(t + s) = 0, t, s ∈ R F xy (t + s) = 1 F xy (t) = 1 F yz (t) = 1 F xz (t + s) = 1 t, s ∈ R (X, F)  (X, F) d : X × X → G X × X G d(x, y)(t) = 1 − F xy (t), t ≥ 0 d(x, y)(t) = 0 t < 0. (X, d, L, R) L R L(a, b) = 0 a, b ∈ [0, 1] R(a, b) =  0 1 d d(x, y) = 0 d(x, y)(t) = 0, t > 0 F xy (t) = 1, t > 0 x = y. x = y F xy (t) = 1 t > 0 d(x, y)(t) = 0 t > 0 F xy (0) = 0 d(x, y)(0) = 1 d d(x, y ) = 0. d(x, y) = 0 x = y. d(x, y) = d(y, x), x, y ∈ X. x, y, z ∈ X L(a, b) = 0 a, b ∈ [0, 1] d(x, y)(t + s)  0 = L(d(x, z)(t), d(z, y)(s)), s, t. d(x, y)(t + s)  L(d(x, z)(t), d(z, y)(s)), s  λ 1 (z, y), t  λ 1 (x, z) s + t  λ 1 (x, y). s  λ 1 (z, y), t  λ 1 (x, z) s + t  λ 1 (x, y) d(x, z)(t) > 0 d(z, y)(s) > 0 d(x, y)(t + s) > 0 R(a, b) =  0 1 d(x, y)(t + s)  R(d(x, z)(t), d(z, y)(s)), s  λ 1 (z, y), t  λ 1 (x, z) s + t  λ 1 (x, y). (X, d, L, R)  (X, F, ∆) d : X × X → G X × X G d(x, y)(t) =  0 t < t xy 1 − F xy (t) t  t xy t xy = sup{t|F xy (t) = 0} (X, d, L, R) L, R L(a, b) = 0 R(a, b) = 1 − ∆(1 − a, 1 − b), a, b ∈ [0, 1]. (X, d, L, R) F xy d(x, y) ∈ G x, y ∈ X L, R L, R : [0, 1]×[0, 1] → [0, 1] x y L(0, 0) = 0 R(1, 1) = 1 d(x, y) = 0 d(x, y)(t) = 0, t = 0 d(x, y)(0) = 1 F xy (t) = 1, t > 0 0 = t xy x = y x = y F xy (t) = 1 t > 0 t = 0 F xy (0) = 0 t xy = 0 d(x, y)(t) = 0 t = 0 d(x, y)(0) = 1 d(x, y) = 0 x = y. x, y ∈ X t ∈ R F F xy (t) = F yx (t) d(x, y) = d(y, x), x, y ∈ X. x, y ∈ X d(x, y)(t + s)  0 = L  d(x, z)(t), d(z, y)(t)  , t, s ∈ R. x, y, z ∈ X t, s  t xy ≥ 0 ∆(1 − d(x, z)(t), 1 − d(z, y )(s)) = ∆(F xz (t), F zy (s))  F xy (t + s) ⇐⇒ 1 − ∆(1 − d(x, z)(t), 1 − d(z, y)(s))  1 − F xy (t + s) ⇐⇒ d(x, y)(t + s)  R(d(x, z)(t), d(z, y)(s)), x, y ∈ X. s  λ 1 (z, y), t  λ 1 (x, z) s + t  λ 1 (x, y)  t xy d(x, y)(t + s)  R(d(x, z)(t), d(z, y)(s)). (X, d, L, R)  (X, d, L, R) lim t→+∞ d(x, y)(t) = 0, x, y ∈ X R(1, a) = R(a, 1) = 1, a ∈ [0, 1] F F xy (t) =  0 t < λ 1 (x, y), 1 − d(x, y)(t) t  λ 1 (x, y). ∆ ∆(a, b) = 1 − R(1 − a, 1 − b) a, b ∈ [0, 1] (X, F, ∆) ∆ t d(x, y) ∈ G d(x, y)(t) F xy F xy F xy F xy (t) = 1 t > 0 d(x, y)(t) = 0 t > 0 d(x, y)(0) = 1 d(x, y)(t) = 0 t < 0 d(x, y) = 0 x = y F xy (t) = 1 t > 0 x = y. F xy (t) = F yx (t) t ∈ R x, y, z ∈ X s, t  0 d, R ∆ F xy (t + s) = 1 − d(x, y)(t + s)  1 − R  d(x, z)(t), d(z, y)(s)  = 1 − R  1 − (1 − d(x, z)(t)), 1 − (1 − d(z, y)(s))  = 1 − R  1 − F xz (t), 1 − F zy (s)  = ∆  F xz (t), F zy (s)  . F xy (t + s)  ∆  F xz (t), F zy (s)  x, y, z ∈ X s, t  0 (X, F, ∆)  124 8 12 α