Подстраница "Користувач:Галактион/Кон'юнкція" создана для того, чтобы перенести информацию из раздела "Обговорення" статьи "Кон'юнкція ". Галактион 18:28, 5 березня 2010 (UTC)
Я не владею Украинским языком, поэтому дополню статью "Конъюнкция" некоторыми сведениями на Русском и Английском языках.
Определение валидности (достоверности) предложения
a
∧
b
{\displaystyle ~\mathrm {a\ \land \ b} }
∧
{\displaystyle ~\land }
ред.
Если известны валидность
V
a
l
i
d
i
t
y
(
a
)
{\displaystyle ~\mathrm {V_{alidity}(a)} }
a
{\displaystyle ~\mathrm {a} }
V
a
l
i
d
i
t
y
(
b
)
{\displaystyle ~\mathrm {V_{alidity}(b)} }
b
{\displaystyle ~\mathrm {b} }
V
a
l
i
d
i
t
y
(
a
∧
b
)
{\displaystyle ~\mathrm {V_{alidity}(a\ \land \ b)} }
a
∧
b
{\displaystyle ~\mathrm {a\ \land \ b} }
Первый способ
⊢
V
1
(
a
)
∈
[
0
,
1
]
∧
V
1
(
b
)
∈
[
0
,
1
]
→
V
1
(
a
∧
b
)
=
V
1
(
a
)
⋅
V
1
(
b
)
{\displaystyle ~\vdash \quad \mathrm {V_{1}(a)\in [0,1]\ \land \ V_{1}(b)\in [0,1]\ \to \ V_{1}(a\ \land \ b)=V_{1}(a)\cdot V_{1}(b)} }
Примеры
⊢
V
1
(
a
)
=
0
∧
V
1
(
b
)
=
0
→
V
1
(
a
∧
b
)
=
V
1
(
a
)
⋅
V
1
(
b
)
=
0
⋅
0
=
0
{\displaystyle ~\vdash \quad \mathrm {V_{1}(a)=0\ \land \ V_{1}(b)=0\quad \to \quad V_{1}(a\ \land \ b)=V_{1}(a)\cdot V_{1}(b)=0\cdot 0=0} }
⊢
V
1
(
a
)
=
1
∧
V
1
(
b
)
=
0
→
V
1
(
a
∧
b
)
=
1
⋅
0
=
0
{\displaystyle ~\vdash \quad \mathrm {V_{1}(a)=1\ \land \ V_{1}(b)=0\quad \to \quad V_{1}(a\ \land \ b)=1\cdot 0=0} }
⊢
V
1
(
a
)
=
0
∧
V
1
(
b
)
=
1
→
V
1
(
a
∧
b
)
=
0
⋅
1
=
0
{\displaystyle ~\vdash \quad \mathrm {V_{1}(a)=0\ \land \ V_{1}(b)=1\quad \to \quad V_{1}(a\ \land \ b)=0\cdot 1=0} }
⊢
V
1
(
a
)
=
1
∧
V
1
(
b
)
=
1
→
V
1
(
a
∧
b
)
=
1
⋅
1
=
1
{\displaystyle ~\vdash \quad \mathrm {V_{1}(a)=1\ \land \ V_{1}(b)=1\quad \to \quad V_{1}(a\ \land \ b)=1\cdot 1=1} }
Примечание
⊢
V
1
(
a
)
∈
[
0
,
1
]
∧
V
1
(
b
)
∈
[
0
,
1
]
→
V
1
(
a
∧
b
)
∈
[
0
,
1
]
{\displaystyle ~\vdash \quad \mathrm {V_{1}(a)\in [0,1]\ \land \ V_{1}(b)\in [0,1]\quad \to \quad V_{1}(a\ \land \ b)\in [0,1]} }
Второй способ
⊢
V
(
a
)
∈
[
0
,
1
]
∧
V
(
b
)
∈
[
0
,
1
]
→
V
(
a
∧
b
)
=
1
2
⋅
(
V
(
a
)
+
V
(
b
)
−
|
V
(
a
)
−
V
(
b
)
|
)
=
=
m
i
n
(
V
(
a
)
,
V
(
b
)
)
{\displaystyle {\begin{aligned}\vdash \quad \mathrm {V(a)\in [0,1]\ \land \ V(b)\in [0,1]\ \to \ V(a\ \land \ b)={\frac {1}{2}}\cdot (V(a)+V(b)-|V(a)-V(b)|)=} \\\ \mathrm {=min(V(a),\ V(b))} \end{aligned}}}
Примеры
⊢
V
(
a
)
=
0
∧
V
(
b
)
=
0
→
V
(
a
∧
b
)
=
m
i
n
(
V
(
a
)
,
V
(
b
)
)
=
m
i
n
(
0
,
0
)
=
0
{\displaystyle ~\vdash \quad \mathrm {V(a)=0\ \land \ V(b)=0\quad \to \quad V(a\ \land \ b)=min(V(a),\ V(b))=min(0,0)=0} }
⊢
V
(
a
)
=
1
∧
V
(
b
)
=
0
→
V
(
a
∧
b
)
=
m
i
n
(
1
,
0
)
=
0
{\displaystyle ~\vdash \quad \mathrm {V(a)=1\ \land \ V(b)=0\quad \to \quad V(a\ \land \ b)=min(1,0)=0} }
⊢
V
(
a
)
=
0
∧
V
(
b
)
=
1
→
V
(
a
∧
b
)
=
m
i
n
(
0
,
1
)
=
0
{\displaystyle ~\vdash \quad \mathrm {V(a)=0\ \land \ V(b)=1\quad \to \quad V(a\ \land \ b)=min(0,1)=0} }
⊢
V
(
a
)
=
1
∧
V
(
b
)
=
1
→
V
(
a
∧
b
)
=
m
i
n
(
1
,
1
)
=
1
{\displaystyle ~\vdash \quad \mathrm {V(a)=1\ \land \ V(b)=1\quad \to \quad V(a\ \land \ b)=min(1,1)=1} }
Примечание
⊢
V
(
a
)
∈
[
0
,
1
]
∧
V
(
b
)
∈
[
0
,
1
]
→
V
(
a
∧
b
)
∈
[
0
,
1
]
{\displaystyle ~\vdash \quad \mathrm {V(a)\in [0,1]\ \land \ V(b)\in [0,1]\quad \to \quad V(a\ \land \ b)\in [0,1]} }
Дополнение
⊢
V
1
(
a
)
∈
[
0
,
1
]
∧
V
(
a
)
∈
[
0
,
1
]
∧
V
1
(
b
)
∈
[
0
,
1
]
∧
V
(
b
)
∈
[
0
,
1
]
∧
V
1
(
a
)
=
V
(
a
)
∧
V
1
(
b
)
=
V
(
b
)
→
V
1
(
a
∧
b
)
≤
V
(
a
∧
b
)
{\displaystyle {\begin{aligned}\mathrm {\vdash \quad V_{1}(a)\in [0,1]\ \land \ V(a)\in [0,1]\quad \land \quad V_{1}(b)\in [0,1]\ \land \ V(b)\in [0,1]\ \land } \\\ \mathrm {V_{1}(a)=V(a)\ \land \ V_{1}(b)=V(b)\quad \to \quad V_{1}(a\ \land \ b)\leq V(a\ \land \ b)} \end{aligned}}}
⊢
V
1
(
a
)
∈
{
0
,
1
}
∧
V
(
a
)
∈
{
0
,
1
}
∧
V
1
(
b
)
∈
{
0
,
1
}
∧
V
(
b
)
∈
{
0
,
1
}
∧
V
1
(
a
)
=
V
(
a
)
∧
V
1
(
b
)
=
V
(
b
)
→
V
1
(
a
∧
b
)
=
V
(
a
∧
b
)
{\displaystyle {\begin{aligned}\vdash \quad \mathrm {V_{1}(a)\in \{0,1\}\ \land \ V(a)\in \{0,1\}\quad \land \quad V_{1}(b)\in \{0,1\}\ \land \ V(b)\in \{0,1\}\ \land } \\\ \mathrm {V_{1}(a)=V(a)\ \land \ V_{1}(b)=V(b)\quad \to \quad V_{1}(a\ \land \ b)=V(a\ \land \ b)} \end{aligned}}}
⊢
V
a
l
i
d
i
t
y
(
T
)
=
1
→
V
a
l
i
d
i
t
y
(
a
∧
T
)
=
V
a
l
i
d
i
t
y
(
T
∧
a
)
=
V
a
l
i
d
i
t
y
(
a
)
≤
1
{\displaystyle ~\vdash \quad \mathrm {V_{alidity}({\mathfrak {T}})=1\ \to \ V_{alidity}(a\ \land \ {\mathfrak {T}})=V_{alidity}({\mathfrak {T}}\ \land \ a)=V_{alidity}(a)\leq 1} }
⊢
V
a
l
i
d
i
t
y
(
f
)
=
0
→
V
a
l
i
d
i
t
y
(
a
∧
f
)
=
V
a
l
i
d
i
t
y
(
f
∧
a
)
=
0
≤
V
a
l
i
d
i
t
y
(
a
)
{\displaystyle ~\vdash \quad \mathrm {V_{alidity}({\mathfrak {f}})=0\ \to \ V_{alidity}(a\ \land \ {\mathfrak {f}})=V_{alidity}({\mathfrak {f}}\ \land \ a)=0\leq V_{alidity}(a)} }
Правила употребления союза
∧
{\displaystyle ~\land }
ред.
Критика, мнения, ...
ред.
![{\displaystyle ~\vdash \quad \mathrm {V_{1}(a)\in [0,1]\ \land \ V_{1}(b)\in [0,1]\ \to \ V_{1}(a\ \land \ b)=V_{1}(a)\cdot V_{1}(b)} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/54f2bc16b7ce749a433e4f623ae91b91e5599a69)
![{\displaystyle ~\vdash \quad \mathrm {V_{1}(a)\in [0,1]\ \land \ V_{1}(b)\in [0,1]\quad \to \quad V_{1}(a\ \land \ b)\in [0,1]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/60a9ad91e2064d8d3c2aa364c8ac4115fe994a10)
![{\displaystyle {\begin{aligned}\vdash \quad \mathrm {V(a)\in [0,1]\ \land \ V(b)\in [0,1]\ \to \ V(a\ \land \ b)={\frac {1}{2}}\cdot (V(a)+V(b)-|V(a)-V(b)|)=} \\\ \mathrm {=min(V(a),\ V(b))} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/643bc9347f2372815bfbcf279555b0e9155bf931)
![{\displaystyle ~\vdash \quad \mathrm {V(a)\in [0,1]\ \land \ V(b)\in [0,1]\quad \to \quad V(a\ \land \ b)\in [0,1]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b69e8cd06a1fecd28635ffdcc11b547058d6294d)
![{\displaystyle {\begin{aligned}\mathrm {\vdash \quad V_{1}(a)\in [0,1]\ \land \ V(a)\in [0,1]\quad \land \quad V_{1}(b)\in [0,1]\ \land \ V(b)\in [0,1]\ \land } \\\ \mathrm {V_{1}(a)=V(a)\ \land \ V_{1}(b)=V(b)\quad \to \quad V_{1}(a\ \land \ b)\leq V(a\ \land \ b)} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2c7f7e26e2d313d07d894158301214ceaf741f)
