简单性质 (1)对任意公式a和指派,a∈{0,1} (2)(3)7和(3→)的值如下表: a|0(a→6) 00 01 10 011 0 1101 与、→的真值表相同
(1) α σ, ασ ∈ {0, 1}. (2) (¬β)σ (β→γ)σ : ασ (¬ α)σ 0 1 1 0 ασ βσ (α→β)σ 0 0 1 0 1 1 1 0 0 1 1 1 ¬ ✓→ ✒ 5
aVβ的值 (1)a∨β是(-a)→β的简写 (aV)0 =((-a)→) =maX{1-(-a),} =maX{1-(1-a0),} 10 =maX{Qa,3”}
α ∨ β (1) α ∨ β (¬ α)→β . (2) (α ∨ β)σ = ((¬ α)→β)σ = max{1 − (¬α)σ, βσ} = max{1 − (1 − ασ), βσ} = max{ασ, βσ} ασ βσ (α ∨ β)σ 0 0 0 0 1 1 1 0 1 1 1 1 6
a∧8的值 (1)a∧β是-(a→-)的简写 (2) (aA)0 a0(a∧B) =1-ma×{1-a,()%}000 =1-max{1-a0,1-60} 01 0 =1-(1+max{-a,-6} =-(-min{a",60}) mina, Boj
α ∧ β (1) α ∧ β ¬ (α → ¬ β) . (2) (α ∧ β)σ = 1 − (α → ¬ β)σ = 1 − max{1 − ασ, (¬β)σ} = 1 − max{1 − ασ, 1 − βσ} = 1 − (1 + max{−ασ, −βσ} = −(− min{ασ, βσ}) = min{ασ, βσ} ασ βσ (α ∧ β)σ 0 0 0 0 1 0 1 0 0 1 1 1 7
a←β的值 (1)aφβ是-((a→B)→-()→a)的简写 (2)(a4B) =1-(a→6)→-(→a)0 =1-max{1-(a→),1-(→a)?} =min{(a→),(→a)} min max 1-a, B01, max1-B0a1 min max1,a+30)-ao max{1,a+}-B0} =maX{1,Q+60}+min{-a0,-B} max(1, a+Bo-maxa, Bo)
α ↔ β (1) α ↔ β ¬((α→β) → ¬(β→α)) . (2) (α ↔ β)σ = 1 − ((α→β) → ¬(β→α))σ = 1 − max{1 − (α→β)σ, 1 − (β→α))σ} = min{(α→β)σ, (β→α)σ} = min{max{1 − ασ, βσ}, max{1 − βσασ}} = min{max{1, ασ + βσ} − ασ, max{1, ασ + βσ} − βσ} = max{1, ασ + βσ} + min{−ασ, −βσ} = max{1, ασ + βσ} − max{ασ, βσ} 8
a←B的值表 a/0(a) 0 0101 1001
α ↔ β ασ βσ (α ↔ β)σ 0 0 1 0 1 0 1 0 0 1 1 1 9