# 【集合论】有序对 ( 有序对 | 有序三元组 | 有序 n 元祖 )

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

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

1. 引理 1 : { x , a } = { x , b } \{ x , a \} = \{ x, b \}{x,a}={x,b} ⇔ \Leftrightarrow⇔ a = b a=ba=b

2. 引理 2 : 若 A = B ≠ ∅ \mathscr{A} = \mathscr{B} \not= \varnothingA=B

=∅ , 则有

① ⋃ A = ⋃ B \bigcup \mathscr{A} = \bigcup \mathscr{B}⋃A=⋃B

② ⋂ A = ⋂ B \bigcap \mathscr{A} = \bigcap \mathscr{B}⋂A=⋂B

3. 定理 : < a , b > = < c , d > <a,b> = <c, d><a,b>=<c,d> ⇔ \Leftrightarrow⇔ a = c ∧ b = d a = c \land b = da=c∧b=d

4. 推论 : a ≠ b a \not= ba

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

=<b,a>

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

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

< 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 >

<a

1

,a

2

,⋯,a

n

>=<<a

1

,⋯,a

n−1

>,a

n

>

n

在后 , 构成有序对 ;

< 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><a

1

,a

2

,⋯,a

n

>=<b

1

,b

2

,⋯,b

n

> ⇔ \Leftrightarrow⇔ a i = b i , i = 1 , 2 , ⋯   , n a_i = b_i , i = 1, 2, \cdots , na

i

=b

i

,i=1,2,⋯,n

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