Definition (A B(AB)(1878)) A and B are equipotent if there exists a bijection from A to B. Q:Is“≈”an equivalence relation? Theorem (The "Equivalence Concept"of Equipotent) For any sets A,B,C: (a)A≈B (b)A≈B→BA (C)A≈BAB≈C→A≈C Hengfeng Wei (hfwei&inju.edu.cn) 1-11 Set Theory (IV):Infinity 2019年12月17日16/49
Definition (|A| = |B| (A ≈ B) (1878)) A and B are equipotent if there exists a bijection from A to B. Q : Is “≈” an equivalence relation? Theorem (The “Equivalence Concept” of Equipotent) For any sets A, B, C: (a) A ≈ B (b) A ≈ B =⇒ B ≈ A (c) A ≈ B ∧ B ≈ C =⇒ A ≈ C Hengfeng Wei (hfwei@nju.edu.cn) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 16 / 49
Definition (Finite) X is finite if 3n EN X n. 1X1=l{0,1,…,n-1H Theorem(UD Theorem 22.6) Let A be a finite set.There is a unique n EN such that A≈{0,1，…，n-1} Hengfeng Wei (hfweiinju.edu.cn)1-11 Set Theory (IV):Infinity 2019年12月17日17/49
Definition (Finite) X is finite if ∃n ∈ N : |X| = n. |X| = |{0, 1, · · · , n − 1}| Theorem (UD Theorem 22.6) Let A be a finite set. There is a unique n ∈ N such that A ≈ {0, 1, · · · , n − 1}. Hengfeng Wei (hfwei@nju.edu.cn) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 17 / 49