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} Jun Ma (majungnju.edu.cn)1-9 Set Theory (II):Relations 2021年12月02日9/52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (Relations) A relation R from A to B is a subset of A × B: R ⊆ A × B Examples I Both A × B and ∅ are relations from A to B. I < = {(a, b) ∈ R × R | a is less than b} I D = {(a, b) ∈ N × N | ∃q ∈ N : a · q = b} I 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} Jun Ma (majun@nju.edu.cn) 1-9 Set Theory (II): Relations 2021 年 12 月 02 日 9 / 52
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|3q∈N:a·q=b} Jun Ma (majungnju.edu.cn)1-9 Set Theory (II):Relations 2021年12月02日9/52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (Relations) A relation R from A to B is a subset of A × B: R ⊆ A × B Examples I Both A × B and ∅ are relations from A to B. I < = {(a, b) ∈ R × R | a is less than b} I D = {(a, b) ∈ N × N | ∃q ∈ N : a · q = b} I 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} Jun Ma (majun@nju.edu.cn) 1-9 Set Theory (II): Relations 2021 年 12 月 02 日 9 / 52
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|3q∈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} Jun Ma (majunginju.edu.cn)1-9 Set Theory (II):Relations 2021年12月02日 9/52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (Relations) A relation R from A to B is a subset of A × B: R ⊆ A × B Examples I Both A × B and ∅ are relations from A to B. I < = {(a, b) ∈ R × R | a is less than b} I D = {(a, b) ∈ N × N | ∃q ∈ N : a · q = b} I 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} Jun Ma (majun@nju.edu.cn) 1-9 Set Theory (II): Relations 2021 年 12 月 02 日 9 / 52
Important Relations: ALL ANIMALS ARE EQUAL BIT SOME A RE MO RE EQUAL TVAN OTVERS Equivalence Relations (1-9) Functions (1-10) Ordering Relations(1-12) 4口，1①卡43，t夏，3)Q0 Jun Na (majunnju.edu.cn1-9 Set Theory (II):Relations 2021年12月02日10/52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Important Relations: Equivalence Relations (1-9) Functions (1-10) Ordering Relations (1-12) Jun Ma (majun@nju.edu.cn) 1-9 Set Theory (II): Relations 2021 年 12 月 02 日 10 / 52
Before that. 3 Definitions 5 Operations 7 Properties R={(0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,3)} 4口·￥①，43，t夏，里Q0 Jun Ma (majunainju.edu.cn) 1-9 Set Theory (II):Relations 2021年12月02日11/52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Definitions 5 Operations 7 Properties R = {(0, 1),(0, 2),(0, 3),(1, 1),(1, 2),(1, 3),(2, 3)} Jun Ma (majun@nju.edu.cn) 1-9 Set Theory (II): Relations 2021 年 12 月 02 日 11 / 52