Definition () x生A≌(x∈A). Definition (C) A二Bx(x∈A→x∈B) 4口￥0,43，t夏里Q0 Juni jtmcmjtedncn Set Theory:Axioms and Operations 2021 11 25 13/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (∈/) x /∈ A , ¬(x ∈ A). Definition (⊆) A ⊆ B , ∀x(x ∈ A =⇒ x ∈ B) Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 13 / 40
ZFC (1)Axiom of Extensionality (2)Axiom of the Empty Set (3)Axiom of Paring (4)Axiom of Union (5)Axiom of Power Set (6)Axiom of Infinity (7)Axiom of Choice (8)Axiom(s)of Separation (9)Axiom(s)of Replacement (10)Axiom of Foundation 4口￥0,3，t夏里Q0 Jun Ma (majunainju.edu.cn) Set Theory:Axioms and Operations2021年11月25日14/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ZFC (1) Axiom of Extensionality (2) Axiom of the Empty Set (3) Axiom of Paring (4) Axiom of Union (5) Axiom of Power Set (6) Axiom of Infinity (7) Axiom of Choice (8) Axiom(s) of Separation (9) Axiom(s) of Replacement (10) Axiom of Foundation Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 14 / 40
ZFC (1)Axiom of Extensionality (2)Axiom of the Empty Set (3)Axiom of Paring (4)Axiom of Union (5)Axiom of Power Set (6)Axiom of Infinity (7)Axiom of Choice (8)Axiom(s)of Separation (9)Axiom(s)of Replacement (10)Axiom of Foundation 4口￥0,3，t夏里Q0 Jun Ma (majunainju.edu.cn) Set Theory:Axioms and Operations2021年11月25日15/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ZFC (1) Axiom of Extensionality (2) Axiom of the Empty Set (3) Axiom of Paring (4) Axiom of Union (5) Axiom of Power Set (6) Axiom of Infinity (7) Axiom of Choice (8) Axiom(s) of Separation (9) Axiom(s) of Replacement (10) Axiom of Foundation Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 15 / 40
Axiom (Axiom of Extensionality) If two sets have exactly the same members,then they are equal. HAB（x(x∈A→x∈B)→A=B) VAB(ACBABCA→A=B): 4口￥0,43，t夏里Q0 Jun 1E (tacmjmcdncn Set Theory:Axioms and Operations 2021 11 25 16/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) . Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 16 / 40
Axiom(Axiom of Extensionality) If two sets have exactly 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夏里Q0 Jun 1E (tacmjmcdncn Set Theory:Axioms and Operations 2021 11 25 16/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) . Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 16 / 40