Definition(Ordered Pairs (Kazimierz Kuratowski;1921)) (a,b){a},{a,b}} Theorem (a,b)=(c,d)→a=c∧b=d Proof. {ah,{a,b}={c,{c,d} CASE I:a=b CASE II:a≠b 4口·￥①，43，t夏里Q0 Hengfeng Wei (hfweiinju.edu.cn)1-9 Set Theory (II):Relations 2019年12月03日7/51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (Ordered Pairs (Kazimierz Kuratowski; 1921)) (a, b) ≜ { {a}, {a, b} } Theorem (a, b) = (c, d) ⇐⇒ a = c ∧ b = d Proof. { {a}, {a, b} } = { {c}, {c, d} } Case I : a = b Case II : a ̸= b Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 7 / 51
Definition(Ordered Pairs(Norbert Wiener;1914)) (a,b){{a,0,{b} *w 4口，1①，43，t夏，3080 Hengfeng Wei (hfweiinju.edu.cn)1-9 Set Theory (II):Relations 2019年12月03日8/51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (Ordered Pairs (Norbert Wiener; 1914)) (a, b) ≜ {{ {a}, ∅ } , { {b} }} Theorem (a, b) = (c, d) ⇐⇒ a = c ∧ b = d Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 8 / 51
Definition(Ordered Pairs(Norbert Wiener;1914)) (a,b){{a,0,{b} pw Theorem (a,b)=(c,d)→a=c∧b=d 4口·￥①，43，t夏里Q0 Hengfeng Wei (hfwei&inju.edu.cn) 1-9 Set Theory (II):Relations 2019年12月03日8/51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (Ordered Pairs (Norbert Wiener; 1914)) (a, b) ≜ {{ {a}, ∅ } , { {b} }} Theorem (a, b) = (c, d) ⇐⇒ a = c ∧ b = d Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 8 / 51
Definition (Cartesian Products) The Cartesian product A x B of A and B is defined as A×Be{(a,b)|a∈AAb∈B} 4口，￥①，43，t豆，30Q0 Hengfeng Wei (hfweiinju.edu.cn)1-9 Set Theory (II):Relations 2019年12月03日9/51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (Cartesian Products) The Cartesian product A × B of A and B is defined as A × B ≜ {(a, b) | a ∈ A ∧ b ∈ B} X2 ≜ X × X Theorem A × B is a set. Proof. A × B ≜ {(a, b) ∈ ? | a ∈ A ∧ b ∈ B} { {a}, {a, b} } ∈ Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 9 / 51
Definition (Cartesian Products) The Cartesian product A x B of A and B is defined as A×B≌{(a,b)|a∈AAb∈B} X2会X×X 4口，￥①，43，t豆，30Q0 Hengfeng Wei (hfweiinju.edu.cn)1-9 Set Theory (II):Relations 2019年12月03日9/51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (Cartesian Products) The Cartesian product A × B of A and B is defined as A × B ≜ {(a, b) | a ∈ A ∧ b ∈ B} X2 ≜ X × X Theorem A × B is a set. Proof. A × B ≜ {(a, b) ∈ ? | a ∈ A ∧ b ∈ B} { {a}, {a, b} } ∈ Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 9 / 51