At mengdeliar as en grünjleien fääk faan't matematiik . Hat befaadet ham diarmä, hü mengden mäenööder tuuphinge an faanenööder ufhinge. Det spriak faan't mengdeliar woort brükt uun a algebra , analysis , geometrii , topologii an ööder matemaatisk feeg.
Det eegenskap
∈
{\displaystyle \in }
(as element faan ) as det wichtagst ferbinjang tesken mengden. Ales, wat diarütj fulagt, san loogisk slütjer.
A
{\displaystyle A}
as en dialmengde faan
B
{\displaystyle B}
Gemiansoom mengde faan
A
{\displaystyle A}
an
B
{\displaystyle B}
Ferianagt mengde faan
A
{\displaystyle A}
an
B
{\displaystyle B}
Sümeetrisk diferens faan
A
{\displaystyle A}
an
B
{\displaystyle B}
Aptäälang
At mengde mä a elementen
x
1
{\displaystyle x_{1}}
bit
x
n
{\displaystyle x_{n}}
häält det element
a
{\displaystyle a}
genau do, wan
a
{\displaystyle a}
mä ian element
x
k
{\displaystyle x_{k}}
auerianstemet:
a
∈
{
x
1
,
…
,
x
n
}
:⟺
a
=
x
1
∨
a
=
x
2
∨
⋯
∨
a
=
x
n
{\displaystyle a\in \{x_{1},\dotsc ,x_{n}\}:\Longleftrightarrow a=x_{1}\vee a=x_{2}\vee \dotsb \vee a=x_{n}}
Det ütjsaag
a
∈
{
1
,
2
,
3
,
4
}
{\displaystyle a\in \{1,2,3,4\}}
as likwäärdag (ekwiwalent) mä:
a
=
1
∨
a
=
2
∨
a
=
3
∨
a
=
4.
{\displaystyle a=1\vee a=2\vee a=3\vee a=4.}
Beskriiwang
At mengde faan
x
{\displaystyle x}
, huarför det eegenskap
P
(
x
)
{\displaystyle P(x)}
tudraapt, häält en element
a
{\displaystyle a}
genau do, wan det eegenskap üüb
a
{\displaystyle a}
tudraapt:
a
∈
{
x
∣
P
(
x
)
}
:⟺
P
(
a
)
{\displaystyle a\in \{x\mid P(x)\}:\Longleftrightarrow P(a)}
Uun a ütjsaagenloogik het det:
∀
x
(
x
∈
A
↔
P
(
x
)
)
{\displaystyle \forall x(x\in A\leftrightarrow P(x))}
Dialmengde
En mengde
A
{\displaystyle A}
as en dialmengde faan
B
{\displaystyle B}
, wan arke element faan
A
{\displaystyle A}
uk element faan
B
{\displaystyle B}
as:
A
⊆
B
:⟺
∀
x
(
x
∈
A
→
x
∈
B
)
{\displaystyle A\subseteq B\ :\Longleftrightarrow \forall x\left(x\in A\,\rightarrow x\in B\right)}
Leesag mengde
En mengde saner elementen as det leesag mengde. Hat woort mä
∅
{\displaystyle \emptyset }
of uk
{
}
{\displaystyle \{\}}
betiakent.
A
=
∅
:⟺
∀
x
(
¬
x
∈
A
)
{\displaystyle A=\emptyset :\Longleftrightarrow \forall x(\neg \,x\in A)}
För det jindial
¬
x
∈
A
{\displaystyle \neg \,x\in A}
täält uk:
x
∉
A
{\displaystyle x\notin A}
.
Gemiansoom mengde
Diar san
U
{\displaystyle U}
mengden. Det gemiansoom mengde faan
U
{\displaystyle U}
as det mengde faan objekten, diar uun arke elementmengde faan
U
{\displaystyle U}
banen san:
⋂
U
:=
{
x
∣
∀
a
∈
U
:
x
∈
a
}
{\displaystyle \bigcap U:=\{x\mid \forall a\in U\colon \;x\in a\}}
Ferianagt mengde
Det ferianagt mengde faan
U
{\displaystyle U}
mengden as det mengde faan objekten, diar uun tumanst ian elemntmengde faan
U
{\displaystyle U}
banen san:
⋃
U
:=
{
x
∣
∃
a
∈
U
:
x
∈
a
}
{\displaystyle \bigcup U:=\{x\mid \exists a\in U\colon \;x\in a\}}
Likedenang mengden
Tau mengden san likdenang, wan diar josalew elementen banen san:
A
=
B
:⟺
∀
x
(
x
∈
A
↔
x
∈
B
)
{\displaystyle A=B:\Longleftrightarrow \forall x\left(x\in A\,\leftrightarrow x\in B\right)}
Diferens an komplement
A
{\displaystyle A}
saner
B
{\displaystyle B}
At diferens of uk restmengde faan
A
{\displaystyle A}
an
B
{\displaystyle B}
("A saner B ") as det mengde faan elementen, diar uun
A
{\displaystyle A}
, oober ei uun
B
{\displaystyle B}
banen san:
A
∖
B
:=
{
x
∣
(
x
∈
A
)
∧
(
x
∉
B
)
}
{\displaystyle A\setminus B:=\{x\mid \left(x\in A\right)\land \left(x\not \in B\right)\}}
Det diferens as uk det komplement faan B uun A :
B
∁
:=
{
x
∣
x
∉
B
}
{\displaystyle B^{\complement }:=\{x\mid x\not \in B\}}
Sümeetrisk diferens
Det as det jindial faan't 'gemiansoom mengde':
A
△
B
:=
{
x
∣
(
x
∈
A
∧
x
∉
B
)
∨
(
x
∈
B
∧
x
∉
A
)
}
=
(
A
∪
B
)
∖
(
A
∩
B
)
=
(
A
∖
B
)
∪
(
B
∖
A
)
{\displaystyle A\triangle B:=\{x\mid \left(x\in A\,\land \,x\not \in B\right)\lor \left(x\in B\,\land \,x\not \in A\right)\}=(A\cup B)\setminus (A\cap B)=(A\setminus B)\cup (B\setminus A)}
Potensmengde
At potensmengde
P
(
A
)
{\displaystyle {\mathcal {P}}(A)}
faan en mengde
A
{\displaystyle A}
as det mengde faan aal a 'dialmengden' faan
A
{\displaystyle A}
.
P
(
A
)
:=
{
X
∣
X
⊆
A
}
{\displaystyle {\mathcal {P}}(A):=\{X\mid X\subseteq A\}}
Karteesisk produkt faan
A
{\displaystyle A}
an
B
{\displaystyle B}
Karteesisk produkt
At produktmengde faan
A
{\displaystyle A}
an
B
{\displaystyle B}
as det mengde faan aal a paaren mä det iarst element ütj
A
{\displaystyle A}
an det ööder ütj
B
{\displaystyle B}
:.
A
×
B
:=
{
(
a
,
b
)
∣
a
∈
A
,
b
∈
B
}
{\displaystyle A\times B:=\left\{(a,b)\mid a\in A,b\in B\right\}}
(1) För dialmengden
A
,
B
,
C
⊆
X
{\displaystyle A,B,C\subseteq X}
täält: Jo san
refleksiif:
A
⊆
A
{\displaystyle A\subseteq A}
antisümeetrisk: Ütj
A
⊆
B
{\displaystyle A\subseteq B}
an
B
⊆
A
{\displaystyle B\subseteq A}
fulegt
A
=
B
{\displaystyle A=B}
transitiif: Ütj
A
⊆
B
{\displaystyle A\subseteq B}
an
B
⊆
C
{\displaystyle B\subseteq C}
fulegt
A
⊆
C
{\displaystyle A\subseteq C}
(2) A mengdenoperatsioonen
∩
{\displaystyle \cap }
an
∪
{\displaystyle \cup }
san komutatiif, asotsiatiif an distributiif:
Asotsiatiifgesets:
(
A
∪
B
)
∪
C
=
A
∪
(
B
∪
C
)
{\displaystyle \left(A\cup B\right)\cup C=A\cup \left(B\cup C\right)}
an
(
A
∩
B
)
∩
C
=
A
∩
(
B
∩
C
)
{\displaystyle \left(A\cap B\right)\cap C=A\cap \left(B\cap C\right)}
Komutatiifgesets:
A
∪
B
=
B
∪
A
{\displaystyle A\cup B=B\cup A}
an
A
∩
B
=
B
∩
A
{\displaystyle A\cap B=B\cap A}
Distributiifgesets:
A
∪
(
B
∩
C
)
=
(
A
∪
B
)
∩
(
A
∪
C
)
{\displaystyle A\cup \left(B\cap C\right)=\left(A\cup B\right)\cap \left(A\cup C\right)}
an
A
∩
(
B
∪
C
)
=
(
A
∩
B
)
∪
(
A
∩
C
)
{\displaystyle A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)}
De Morgan sin gesetsen:
(
A
∪
B
)
∁
=
A
∁
∩
B
∁
{\displaystyle \left(A\cup B\right)^{\complement }=A^{\complement }\cap B^{\complement }}
an
(
A
∩
B
)
∁
=
A
∁
∪
B
∁
{\displaystyle \left(A\cap B\right)^{\complement }=A^{\complement }\cup B^{\complement }}
Absorptionsgesets:
A
∪
(
A
∩
B
)
=
A
{\displaystyle A\cup \left(A\cap B\right)=A}
an
A
∩
(
A
∪
B
)
=
A
{\displaystyle A\cap \left(A\cup B\right)=A}
(3) Gesetsen för diferensmengden:
Asotsiatiifgesetsen:
(
A
∖
B
)
∖
C
=
A
∖
(
B
∪
C
)
{\displaystyle (A\setminus B)\setminus C=A\setminus (B\cup C)}
an
A
∖
(
B
∖
C
)
=
(
A
∖
B
)
∪
(
A
∩
C
)
{\displaystyle A\setminus (B\setminus C)=(A\setminus B)\cup (A\cap C)}
Distributiifgesetsen:
(
A
∩
B
)
∖
C
=
(
A
∖
C
)
∩
(
B
∖
C
)
{\displaystyle (A\cap B)\setminus C=(A\setminus C)\cap (B\setminus C)}
an
(
A
∪
B
)
∖
C
=
(
A
∖
C
)
∪
(
B
∖
C
)
{\displaystyle (A\cup B)\setminus C=(A\setminus C)\cup (B\setminus C)}
an
A
∖
(
B
∩
C
)
=
(
A
∖
B
)
∪
(
A
∖
C
)
{\displaystyle A\setminus (B\cap C)=(A\setminus B)\cup (A\setminus C)}
an
A
∖
(
B
∪
C
)
=
(
A
∖
B
)
∩
(
A
∖
C
)
{\displaystyle A\setminus (B\cup C)=(A\setminus B)\cap (A\setminus C)}
(4) Gesetsen för sümeetrisk diferensen:
Asotsiatiifgesets:
(
A
△
B
)
△
C
=
A
△
(
B
△
C
)
{\displaystyle (A\triangle B)\triangle C=A\triangle (B\triangle C)}
Komutatiifgesets:
A
△
B
=
B
△
A
{\displaystyle A\triangle B=B\triangle A}
Distributiifgesets:
(
A
△
B
)
∩
C
=
(
A
∩
C
)
△
(
B
∩
C
)
{\displaystyle (A\triangle B)\cap C=(A\cap C)\triangle (B\cap C)}
A
△
∅
=
A
{\displaystyle A\triangle \emptyset =A\quad }
an
A
△
A
=
∅
{\displaystyle A\triangle A=\emptyset }
Wikibooks: Mengdeliar (sjiisk)
Wikibooks: Bewis uun a mengdeliar (sjiisk)