[set theory] ordered pair (ordered pair | ordered triple | ordered n ancestor)

One 、 Ordered pair

---

Ordered pair concept :

<

a

,

b

>

=

{

{

a

}

,

{

a

,

b

}

}

<a, b> = \{ \{ a \} , \{ a , b \} \}

<a,b>={ { a},{ a,b}}

among

a

a

a It's the first element ,

b

b

b It's the second element ;

Remember to do

<

a

,

b

>

<a, b>

<a,b> , You can also remember to do

(

a

,

b

)

(a , b)

(a,b)

understand 1 :

a

,

b

a, b

a,b There is a sequence , The element in the collection of single elements is the first element , The other element in the set of two elements is the second element ;

understand 2 ( recommend ) : The first element appears in each subset , The second element only appears in a subset , In this way , Ensures the definition of ordered pairs , Two elements before and after , The sequence is different , The corresponding ordered pairs are different ;

Here are the different ordered pairs of the same two elements :

Ordered pair

<

a

,

b

>

=

{

{

a

}

,

{

a

,

b

}

}

<a, b> = \{ \{ a \} , \{ a , b \} \}

<a,b>={ { a},{ a,b}}

Ordered pair

<

b

,

a

>

=

{

{

b

}

,

{

a

,

b

}

}

<b, a> = \{ \{ b \} , \{ a , b \} \}

<b,a>={ { b},{ a,b}}

Two 、 Lemma of ordered pair properties 、 Theorem

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1. lemma 1 :

{

x

,

a

}

=

{

x

,

b

}

\{ x , a \} = \{ x, b \}

{ x,a}={ x,b}

⇔

\Leftrightarrow

⇔

a

=

b

a=b

a=b

If two sets are equal , If and only if

a

=

b

a = b

a=b ;

2. lemma 2 : if

A

=

B

≠

∅

\mathscr{A} = \mathscr{B} \not= \varnothing

A=B​=∅ , Then there are

①

⋃

A

=

⋃

B

\bigcup \mathscr{A} = \bigcup \mathscr{B}

⋃A=⋃B

②

⋂

A

=

⋂

B

\bigcap \mathscr{A} = \bigcap \mathscr{B}

⋂A=⋂B

explain : Set family

A

\mathscr{A}

A And Set family

B

\mathscr{B}

B equal , also Neither set family is empty , that The generalized intersection of two set families is equal , The generalized union of two set families is also equal ;

3. Theorem :

<

a

,

b

>

=

<

c

,

d

>

<a,b> = <c, d>

<a,b>=<c,d>

⇔

\Leftrightarrow

⇔

a

=

c

∧

b

=

d

a = c \land b = d

a=c∧b=d

Through the above theorem , It shows that ordered pairs are ordered ;

4. inference :

a

≠

b

a \not= b

a​=b

⇒

\Rightarrow

⇒

<

a

,

b

>

≠

<

b

,

a

>

<a,b> \not= <b, a>

<a,b>​=<b,a>

3、 ... and 、 Ordered triples

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Ordered triples :

<

a

,

b

,

c

>

=

<

<

a

,

b

>

,

c

>

<a, b, c> = < <a, b> , c >

<a,b,c>=<<a,b>,c>

Ordered triples are ordered triples first , The third element is after , The order of composition is right ;

Orderly

n

n

n Yuan Zu :

n

≥

2

n \geq 2

n≥2

<

a

1

,

a

2

,

⋯
 

,

a

n

>

=

<

<

a

1

,

⋯
 

,

a

n

−

1

>

,

a

n

>

<a_1, a_2, \cdots , a_n> = < <a_1, \cdots , a_{n-1}> , a_n >

<a1​,a2​,⋯,an​>=<<a1​,⋯,an−1​>,an​>

Take it first

n

−

1

n-1

n−1 Elements form an order

n

−

1

n-1

n−1 Yuan Zu , The

n

−

1

n-1

n−1 Yuanzu was in front , Then follow No

n

n

n Elements

a

n

a_n

an​ After , Form an orderly pair ;

Four 、 Orderly n Tuple property theorem

---

Orderly

n

n

n Tuple property theorem :

<

a

1

,

a

2

,

⋯
 

,

a

n

>

=

<

b

1

,

b

2

,

⋯
 

,

b

n

>

<a_1, a_2, \cdots , a_n> = <b_1, b_2, \cdots , b_n>

<a1​,a2​,⋯,an​>=<b1​,b2​,⋯,bn​>

⇔

\Leftrightarrow

⇔

a

i

=

b

i

,

i

=

1

,

2

,

⋯
 

,

n

a_i = b_i , i = 1, 2, \cdots , n

ai​=bi​,i=1,2,⋯,n

explain : Two in order

n

n

n Yuan Zu , The elements in each corresponding position are the same , Two

n

n

n Tuples are equal only when they are ordered ;