Axiom (Ordered Pairs) (a,b)=(c,d→a=c∧b=d 4口，1①，43，t夏，30Q0 Hengfeng Wei (bhfweionjn.edu.cn 1-9 Set Theory (II):Relations 2019 12 03 6/51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axiom (Ordered Pairs) (a, b) = (c, d) ⇐⇒ a = c ∧ b = d Definition (Ordered Pairs (Kazimierz Kuratowski; 1921)) (a, b) ≜ { {a}, {a, b} } Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 6 / 51
Axiom (Ordered Pairs) (a,b)=(c,d→a=c∧b=d Definition (Ordered Pairs(Kazimierz Kuratowski;1921)) (a,b){a},{a,b} Hengfeng Wei (hfwei&inju.edu.cn) 1-9 Set Theory (II):Relations 2019年12月03日6/51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axiom (Ordered Pairs) (a, b) = (c, d) ⇐⇒ a = c ∧ b = d Definition (Ordered Pairs (Kazimierz Kuratowski; 1921)) (a, b) ≜ { {a}, {a, b} } Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 6 / 51
Definition(Ordered Pairs (Kazimierz Kuratowski;1921)) (a,b)≌{a,{a,b} 4口，1①，43，t夏，30Q0 Hengeng Wei (thfweionjn.edu.cn 1-9 Set Theory (II):Relations 2019 1203 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 (Kazimierz Kuratowski;1921)) (a,b){a},{a,b} Theorem (a,b)=(c,d→a=c∧b=d 4口，1①，43，t夏，30Q0 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 (Kazimierz Kuratowski;1921)) (a,b){a},{a,b} Theorem (a,b)=(c,d→a=c∧b=d Proof. {ah,{a,b}={c,{c,d} 4口，1①卡43，t夏，3Q0 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