当前位置：网站首页>[set theory] order relation (eight special elements in partial order relation | ① maximum element | ② minimum element | ③ maximum element | ④ minimum element | ⑤ upper bound | ⑥ lower bound | ⑦ minimu

[set theory] order relation (eight special elements in partial order relation | ① maximum element | ② minimum element | ③ maximum element | ④ minimum element | ⑤ upper bound | ⑥ lower bound | ⑦ minimu

2022-07-03 09:15:00 【Programmer community】

List of articles

One 、 The biggest dollar
Two 、 Minimum element
3、 ... and 、 The biggest dollar 、 Minimum element example
Four 、 Great dollar
5、 ... and 、 Minima
6、 ... and 、 Great dollar 、 Minimal element example
7、 ... and 、 upper bound
8、 ... and 、 Lower bound
Nine 、 upper bound 、 Lower bound example
Ten 、 Upper bound ( Minimum upper bound )
11、 ... and 、 Lower bound ( Maximum lower bound )
Twelve 、 Upper bound 、 Infimum example

One 、 The biggest dollar

≼

<A, \preccurlyeq>

$< A, ≼ >$ yes Posets ,

⊆

B \subseteq A

$B \subseteq A$ ,

∈

y \in B

$y \in B$ ,

$B$ All elements in are related to

$y$ Are comparable ,

$B$ Any element in

$x$ , All satisfied with

$x$ Less than or equal to

$y$

Symbolize :

∀

(

∈

→

≼

)

\forall x ( x \in B \to x \preccurlyeq y )

$\forall x (x \in B \to x ≼ y)$

call

$y$ yes

$B$ The largest element of a set ;

Two 、 Minimum element

≼

<A, \preccurlyeq>

$< A, ≼ >$ yes Posets ,

⊆

B \subseteq A

$B \subseteq A$ ,

∈

y \in B

$y \in B$ ,

$B$ All elements in are related to

$y$ Are comparable ,

$B$ Any element in

$x$ , All satisfied with

$y$ Less than or equal to

$x$

Symbolize :

∀

(

∈

→

≼

)

\forall x ( x \in B \to y \preccurlyeq x )

$\forall x (x \in B \to y ≼ x)$

call

$y$ yes

$B$ The smallest element of a set ;

3、 ... and 、 The biggest dollar 、 Minimum element example

aggregate

{

}

A = \{ 1, 2, 3, 4, 5, 6, 9, 10, 15 \}

$A = {1, 2, 3, 4, 5, 6, 9, 10, 15}$ ,

aggregate

$A$ On the division relationship “

∣

$∣$ ” It's a partial order relationship ,

Poset is

∣

<A, |>

$< A, ∣ >$

x
x
$x$ to be divisible by
y
y
$y$ ,
x
x
$x$ It's a divisor ( The denominator ) ,
y
y
$y$ It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
$\frac{y}{x}$
y
y
$y$ Can be
x
x
$x$ to be divisible by ,
x
x
$x$ It's a divisor ( The denominator ) ,
y
y
$y$ It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
$\frac{y}{x}$

Draw the above partially ordered set of hastu :

Insert picture description here

{

}

B_1 = \{ 1,2,3 \}

$B_{1} = {1, 2, 3}$

{

}

B_2 = \{ 3 , 5, 15 \}

$B_{2} = {3, 5, 15}$

B_3 = A

$B_{3} = A$

Find the The biggest dollar , Minimum element ?

{

}

B_1 = \{ 1,2,3 \}

$B_{1} = {1, 2, 3}$

The biggest dollar :
Minimum element :

{

}

B_2 = \{ 3 , 5, 15 \}

$B_{2} = {3, 5, 15}$

The biggest dollar :
Minimum element :

{

}

B_3 = A = \{ 1, 2, 3, 4, 5, 6, 9, 10, 15 \}

$B_{3} = A = {1, 2, 3, 4, 5, 6, 9, 10, 15}$

The biggest dollar :
Minimum element :

Four 、 Great dollar

≼

<A, \preccurlyeq>

$< A, ≼ >$ yes Posets ,

⊆

B \subseteq A

$B \subseteq A$ ,

∈

y \in B

$y \in B$ ,

stay

$B$ There is no better than

$y$ The bigger element ,

Symbolize :

∀

(

∈

∧

≼

→

)

\forall x ( x \in B \land y \preccurlyeq x \to x = y )

$\forall x (x \in B \land y ≼ x \to x = y)$

call

$y$ yes

$B$ A collection of Great dollar ;

5、 ... and 、 Minima

≼

<A, \preccurlyeq>

$< A, ≼ >$ yes Posets ,

⊆

B \subseteq A

$B \subseteq A$ ,

∈

y \in B

$y \in B$ ,

stay

$B$ There is no better than

$y$ Smaller elements ,

Symbolize :

∀

(

∈

∧

≼

→

)

\forall x ( x \in B \land x \preccurlyeq y \to x = y )

$\forall x (x \in B \land x ≼ y \to x = y)$

call

$y$ yes

$B$ A collection of Minima ;

6、 ... and 、 Great dollar 、 Minimal element example

aggregate

{

}

A = \{ 1, 2, 3, 4, 5, 6, 9, 10, 15 \}

$A = {1, 2, 3, 4, 5, 6, 9, 10, 15}$ ,

aggregate

$A$ On the division relationship “

∣

$∣$ ” It's a partial order relationship ,

Poset is

∣

<A, |>

$< A, ∣ >$

x
x
$x$ to be divisible by
y
y
$y$ ,
x
x
$x$ It's a divisor ( The denominator ) ,
y
y
$y$ It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
$\frac{y}{x}$
y
y
$y$ Can be
x
x
$x$ to be divisible by ,
x
x
$x$ It's a divisor ( The denominator ) ,
y
y
$y$ It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
$\frac{y}{x}$

Draw the above partially ordered set of hastu :

Insert picture description here

{

}

B_1 = \{ 1,2,3 \}

$B_{1} = {1, 2, 3}$

{

}

B_2 = \{ 3 , 5, 15 \}

$B_{2} = {3, 5, 15}$

B_3 = A

$B_{3} = A$

Find the Great dollar , Minima ?

{

}

B_1 = \{ 1,2,3 \}

$B_{1} = {1, 2, 3}$

Great dollar :
Minima :

{

}

B_2 = \{ 3 , 5, 15 \}

$B_{2} = {3, 5, 15}$

Great dollar :
Minimum element :

{

}

B_3 = A = \{ 1, 2, 3, 4, 5, 6, 9, 10, 15 \}

$B_{3} = A = {1, 2, 3, 4, 5, 6, 9, 10, 15}$

Great dollar :
Minima :

7、 ... and 、 upper bound

≼

<A, \preccurlyeq>

$< A, ≼ >$ yes Posets ,

⊆

B \subseteq A

$B \subseteq A$ ,

∈

y \in A

$y \in A$

$y$ Than

$B$ All elements in should be big

Symbolize :

∀

(

∈

→

≼

)

\forall x ( x \in B \to x \preccurlyeq y )

$\forall x (x \in B \to x ≼ y)$

call

$y$ yes

$B$ A collection of upper bound ;

8、 ... and 、 Lower bound

≼

<A, \preccurlyeq>

$< A, ≼ >$ yes Posets ,

⊆

B \subseteq A

$B \subseteq A$ ,

∈

y \in A

$y \in A$

$y$ Than

$B$ All elements in should be small

Symbolize :

∀

(

∈

→

≼

)

\forall x ( x \in B \to y \preccurlyeq x )

$\forall x (x \in B \to y ≼ x)$

call

$y$ yes

$B$ A collection of Lower bound ;

Nine 、 upper bound 、 Lower bound example

aggregate

{

}

A = \{ 1, 2, 3, 4, 5, 6, 9, 10, 15 \}

$A = {1, 2, 3, 4, 5, 6, 9, 10, 15}$ ,

aggregate

$A$ On the division relationship “

∣

$∣$ ” It's a partial order relationship ,

Poset is

∣

<A, |>

$< A, ∣ >$

x
x
$x$ to be divisible by
y
y
$y$ ,
x
x
$x$ It's a divisor ( The denominator ) ,
y
y
$y$ It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
$\frac{y}{x}$
y
y
$y$ Can be
x
x
$x$ to be divisible by ,
x
x
$x$ It's a divisor ( The denominator ) ,
y
y
$y$ It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
$\frac{y}{x}$

Draw the above partially ordered set of hastu :

Insert picture description here

{

}

B_1 = \{ 1,2,3 \}

$B_{1} = {1, 2, 3}$

{

}

B_2 = \{ 3 , 5, 15 \}

$B_{2} = {3, 5, 15}$

B_3 = A

$B_{3} = A$

Find the upper bound , Lower bound ?

{

}

B_1 = \{ 1,2,3 \}

$B_{1} = {1, 2, 3}$

upper bound : $B_1B1 All elements in the are big , 6 66 It's the upper bound ;$
Lower bound : $B_1B1 All elements in are small , 1 11 It's the lower boundary ;$

{

}

B_2 = \{ 3 , 5, 15 \}

$B_{2} = {3, 5, 15}$

upper bound : $B_2B2 All elements in the are big , 15 1515 It's the upper bound ;$
Lower bound : $B_2B2 All elements in are small , 1 11 It's the lower boundary ;$

{

}

B_3 = A = \{ 1, 2, 3, 4, 5, 6, 9, 10, 15 \}

$B_{3} = A = {1, 2, 3, 4, 5, 6, 9, 10, 15}$

upper bound : There are no elements and $B_3B3 The elements in are comparable ; There is no upper bound ;$
Lower bound : $B_3B3 The elements in are comparable , 1 11 Than B 3 B_3B3 All elements in are small , 1 11 It's the lower boundary ;$

Ten 、 Upper bound ( Minimum upper bound )

≼

<A, \preccurlyeq>

$< A, ≼ >$ yes Posets ,

⊆

B \subseteq A

$B \subseteq A$ ,

∈

y \in A

$y \in A$

The smallest element in the upper bound is Minimum upper bound , Also known as supremum

11、 ... and 、 Lower bound ( Maximum lower bound )

≼

<A, \preccurlyeq>

$< A, ≼ >$ yes Posets ,

⊆

B \subseteq A

$B \subseteq A$ ,

∈

y \in A

$y \in A$

The biggest element in the lower bound is Maximum lower bound , Also known as infimum

Twelve 、 Upper bound 、 Infimum example

aggregate

{

}

A = \{ 1, 2, 3, 4, 5, 6, 9, 10, 15 \}

$A = {1, 2, 3, 4, 5, 6, 9, 10, 15}$ ,

aggregate

$A$ On the division relationship “

∣

$∣$ ” It's a partial order relationship ,

Poset is

∣

<A, |>

$< A, ∣ >$

x
x
$x$ to be divisible by
y
y
$y$ ,
x
x
$x$ It's a divisor ( The denominator ) ,
y
y
$y$ It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
$\frac{y}{x}$
y
y
$y$ Can be
x
x
$x$ to be divisible by ,
x
x
$x$ It's a divisor ( The denominator ) ,
y
y
$y$ It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
$\frac{y}{x}$

Draw the above partially ordered set of hastu :

Insert picture description here

{

}

B_1 = \{ 1,2,3 \}

$B_{1} = {1, 2, 3}$

{

}

B_2 = \{ 3 , 5, 15 \}

$B_{2} = {3, 5, 15}$

B_3 = A

$B_{3} = A$

Find the Upper bound ( Minimum upper bound ) , Lower bound ( Maximum lower bound ) ?

{

}

B_1 = \{ 1,2,3 \}

$B_{1} = {1, 2, 3}$

Upper bound : $B_1B1 All elements in the are big , 6 66 It's the upper bound ; 6 66 It is also the supremum , Minimum upper bound ;$
Lower bound : $B_1B1 All elements in are small , 1 11 It's the lower boundary ; 1 11 It is also the infimum , Maximum lower bound ;$

{

}

B_2 = \{ 3 , 5, 15 \}

$B_{2} = {3, 5, 15}$

Upper bound : $B_2B2 All elements in the are big , 15 1515 It's the upper bound ; 15 1515 It is also the supremum , Minimum upper bound ;$
Lower bound : $B_2B2 All elements in are small , 1 11 It's the lower boundary ; 1 11 It is also the infimum , Maximum lower bound ;$

{

}

B_3 = A = \{ 1, 2, 3, 4, 5, 6, 9, 10, 15 \}

$B_{3} = A = {1, 2, 3, 4, 5, 6, 9, 10, 15}$

Upper bound : There are no elements and $B_3B3 The elements in are comparable ; There is no upper bound ; non-existent Upper bound / Minimum upper bound ;$
Lower bound : $B_3B3 The elements in are comparable , 1 11 Than B 3 B_3B3 All elements in are small , 1 11 It's the lower boundary ; 1 11 It is also the infimum , Maximum lower bound ;$