Definition(Cartesian Products) The Cartesian product A x B of A and B is defined as A×Be{(a,b)|a∈AAb∈B} X2会X×X Theorem A×B is a set.. 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×Be{(a,b)|a∈A∧b∈B} X2会X×X Theorem A×B is a set.. Proof. A×B≌{(a,b)∈？|a∈AAb∈B} 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} } ∈ 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×Be{(a,b)|a∈A∧b∈B} X2会X×X Theorem A×B is a set.. Proof. A×B≌{(a,b)∈？|a∈AAb∈B} {a,{a,b}∈？ 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} } ∈ ? 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 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 4口，￥①卡43，t夏，3Q0 Hengfeng Wei (hfweiinju.cdu.cn1-9 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