Definition (Relations) A relation R from A to B is a subset of A x B: RCAXB Examples Both A x B and 0 are relations from A to B. <={(a,b)∈R×R|a is less than b} D={(a,b)∈N×N|g∈N:a·q=b} P:the set of people M={(a,b)∈P×P|a is the mother of b} B={(a,b)∈P×P|a is the brother of b} Hengfeng Wei (hfweiinju.edu.cn)1-9 Set Theory (II):Relations 2019年12月03日11/51
Definition (Relations) A relation R from A to B is a subset of A × B: R ⊆ A × B Examples ▶ Both A × B and ∅ are relations from A to B. ▶ < = {(a, b) ∈ R × R | a is less than b} ▶ D = {(a, b) ∈ N × N | ∃q ∈ N : a · q = b} ▶ P : the set of people M = {(a, b) ∈ P × P | a is the mother of b} B = {(a, b) ∈ P × P | a is the brother of b} Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 11 / 51
Important Relations: ALL ANIMALS ARE EQUAL BIT SOMEARE MO RE EQHAL TVAN OTVERS Equivalence Relations (1-9) Functions(1-10) Ordering Relations(1-12) Hengfeng Wei (hfweiinju.edu.cn)1-9 Set Theory (II):Relations 2019年12月03日12/51
Important Relations: Equivalence Relations (1-9) Functions (1-10) Ordering Relations (1-12) Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 12 / 51
Before that. 3 Definitions 5 Operations 7 Properties R={(0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,3)} Hengfeng Wei (hfweixinju.edu.cn) 1-9 Set Theory (II):Relations 2019年12月03日13/51
3 Definitions 5 Operations 7 Properties R = {(0, 1),(0, 2),(0, 3),(1, 1),(1, 2),(1, 3),(2, 3)} Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 13 / 51
3 Definitions Hengfeng Wei fweiinju.edu.cn1-9 Set Theory (II:Relations 2019年12月03日14/51
3 Definitions Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 14 / 51
Definition(Domain) dom(R)=fa:(a,b)ER Theorem dom(R)is a set. dom(R)={a∈？UUr13b:(a,b)∈R} (a,b)={a},{a,b}∈R {a,b}ER a∈UUR Hengfeng Wei (hfweixinju.edu.cn) 1-9 Set Theory (II):Relations 2019年12月03日15/51
Definition (Domain) dom(R) = {a | ∃b : (a, b) ∈ R} Theorem dom(R) is a set. dom(R) = {a ∈ ? ∪∪R | ∃b : (a, b) ∈ R} (a, b) = {{a}, {a, b}} ∈ R {a, b} ∈ ∪ R a ∈ ∪∪R Hengfeng Wei (hfwei@nju.edu.cn) 1-9 Set Theory (II): Relations 2019 年 12 月 03 日 15 / 51