Definition () x生A≌(x∈A). Definition (C) ACB≌x(x∈A→x∈B) 4口￥0,43，t里，里Q0 Hengong Wei Chiweinjnedm.cn Set Theory:Axioms and Operations 2019 11 26 13/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (∈/) x /∈ A ≜ ¬(x ∈ A). Definition (⊆) A ⊆ B ≜ ∀x(x ∈ A =⇒ x ∈ B) Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 13 / 38
Axiom (Axiom of Extensionality) If two sets have eractly the same members,then they are equal. HAB（x(x∈A→x∈B)→A=B) VAB(ACBABCA→A=B): 4口￥0,43，t里，里Q0 Hengong We Chiweinjnedm.cn Set Theory:Axioms and Operations 2019 11 26 14/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axiom (Axiom of Extensionality) If two sets have exactly the same members, then they are equal. ∀A ∀B ( ∀x(x ∈ A ⇐⇒ x ∈ B) =⇒ A = B ) . ∀A ∀B ( A ⊆ B ∧ B ⊆ A =⇒ A = B ) . ∀A ∀B ( A ⊆ B ∧ B ⊆ A ⇐⇒ A = B ) . Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 14 / 38
Axiom (Axiom of Extensionality) If two sets have eractly the same members,then they are equal. HAB（x(x∈A→x∈B)→A=B) VAB(ACBABCA→A=B): VAB(ACB∧BCA→A=B): 4口￥0,43，t里，里Q0 Hengong We Chiweinjnedm.cn Set Theory:Axioms and Operations 2019 11 26 14/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axiom (Axiom of Extensionality) If two sets have exactly the same members, then they are equal. ∀A ∀B ( ∀x(x ∈ A ⇐⇒ x ∈ B) =⇒ A = B ) . ∀A ∀B ( A ⊆ B ∧ B ⊆ A =⇒ A = B ) . ∀A ∀B ( A ⊆ B ∧ B ⊆ A ⇐⇒ A = B ) . Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 14 / 38
Axiom (Empty Set Axiom) There is a set having no members: 3Bx(x主B) 4口￥0,43，t里，里Q0 Hengong Wei Chiweinjnedm.cn Set Theory:Axioms and Operations 2019 11 26 15/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axiom (Empty Set Axiom) There is a set having no members: ∃B ∀x(x /∈ B). Theorem (Uniqueness of Empty Set) There is only one empty set. Definition (“∅”) ∅ ≜ the unique unique empty set. Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 15 / 38
Axiom(Empty Set Axiom) There is a set having no members: 3Bx(x主B) Theorem(Uniqueness of Empty Set) There is only one empty set. 4口￥0,43，t里，里Q0 Hengong Wei Chiweinjnedm.cn Set Theory:Axioms and Operations 2019 11 26 15/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axiom (Empty Set Axiom) There is a set having no members: ∃B ∀x(x /∈ B). Theorem (Uniqueness of Empty Set) There is only one empty set. Definition (“∅”) ∅ ≜ the unique unique empty set. Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 15 / 38