IPA : [ ˈprɛdikaːtumkɒlkuluʃ]
predikátumkalkulus
( matematika ) predikátumlogika
¬
(
(
∀
x
)
P
(
x
)
)
≡
(
∃
x
)
¬
P
(
x
)
{\displaystyle \neg ((\forall x)P(x))\equiv (\exists x)\neg P(x)}
¬
(
(
∃
x
)
P
(
x
)
)
≡
(
∀
x
)
¬
P
(
x
)
{\displaystyle \neg ((\exists x)P(x))\equiv (\forall x)\neg P(x)}
Példa:
Az
a
n
{\displaystyle a_{n}}
A
{\displaystyle A}
ε
>
0
{\displaystyle \varepsilon >0}
|
a
n
−
A
|
<
ε
,
{\displaystyle |a_{n}-A|<\varepsilon ,}
n
≥
N
.
{\displaystyle n\geq N.}
P
(
n
,
ε
)
=
|
a
n
−
A
|
<
ε
,
R
(
n
,
N
)
=
n
≥
N
,
{\displaystyle P(n,\varepsilon )=|a_{n}-A|<\varepsilon ,R(n,N)=n\geq N,}
ahol a
P
{\displaystyle P}
R
{\displaystyle R}
n
∈
N
,
ε
>
0
,
N
∈
N
.
{\displaystyle n\in \mathbb {N} ,\varepsilon >0,N\in \mathbb {N} .}
Ekkor az
a
n
→
A
{\displaystyle a_{n}\to A}
(
∀
ε
)
(
∃
N
)
(
∀
n
)
(
R
(
n
,
N
)
→
P
(
n
,
ε
)
)
{\displaystyle (\forall \varepsilon )(\exists N)(\forall n)(\mathbb {R} (n,N)\to P(n,\varepsilon ))}
Azt, hogy
a
n
{\displaystyle a_{n}}
A
{\displaystyle A}
(
∃
ε
)
(
∀
N
)
(
∃
n
)
(
R
(
n
,
N
)
∧
(
¬
P
(
n
,
ε
)
)
)
{\displaystyle (\exists \varepsilon )(\forall N)(\exists n)(\mathbb {R} (n,N)\wedge (\neg P(n,\varepsilon )))}
Azaz létezik olyan
ε
>
0
{\displaystyle \varepsilon >0}
N
{\displaystyle N}
|
a
n
−
A
|
≥
ε
{\displaystyle |an-A|\geq \varepsilon }
n
≥
N
{\displaystyle n\geq N}
