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}} 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 (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} pw Theorem (a,b)=(c,d)→a=c∧b=d 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×B≌{(a,b)|a∈AAb∈B} X2会X×X Theorem A×B is a set.. Proof. A×B≌{(a,b)∈？|a∈AAb∈B} {a,{a,b}∈？P(P(AUB) Hengfeng Wei (hfwei&inju.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} } ∈ ?P(P(A ∪ B)) Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 9 / 51
Definition (Relations) A relation R from A to B is a subset of A x B: RCAXB If A B,R is called a relation on A. Definition (Notations) (a,b)∈R R(a,b) aRb Hengfeng Wei (hfweixinju.edu.cn) 1-9 Set Theory (II):Relations 2019年12月03日10/51
Definition (Relations) A relation R from A to B is a subset of A × B: R ⊆ A × B If A = B, R is called a relation on A. Definition (Notations) (a, b) ∈ R R(a, b) aRb Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 10 / 51