Definition The set of all functions from X to Y: Yx={fIf:X→Y Q:Is there a set consisting of all functions? Theorem There is no set consisting of all functions. Suppose by contradiction that A is the set of all functions. For every set X,there exists a function Ix:. dom(Ix) Ix∈A Hengfeng Wei (hfweinju.edu.cn)1-10 Set Theory (III):Functions 2019年12月10日11/40
Definition The set of all functions from X to Y : Y X = {f | f : X → Y } Q : Is there a set consisting of all functions? Theorem There is no set consisting of all functions. Suppose by contradiction that A is the set of all functions. For every set X, there exists a function IX : {X} → {X}. ∪ IX∈A dom(IX) Hengfeng Wei (hfwei@nju.edu.cn) 1-10 Set Theory (III): Functions 2019 年 12 月 10 日 11 / 40
Functions as Sets Hengfeng Wei (hfweiinju.edu.cn)1-10 Set Theory (III):Functions 2019年12月10日12/40
Functions as Sets Hengfeng Wei (hfwei@nju.edu.cn) 1-10 Set Theory (III): Functions 2019 年 12 月 10 日 12 / 40
Axiom (Axiom of Extensionality) VA VBVx:(x∈A→x∈B)→A=B. Theorem(The Principle of Functional Extensionality) f,g are functions: f=g→dom(f)=dom(g)∧（付x∈dom(f):f(x)=g(x)） fg(a,b):(a,b)∈f→(a,b)∈g)→f=g It may be that cod(f)cod(g). Hengfeng Wei (hfweixinju.edu.cn) 1-10 Set Theory (III):Functions 2019年12月10日13/40
Axiom (Axiom of Extensionality) ∀A ∀B ∀x : (x ∈ A ⇐⇒ x ∈ B) ⇐⇒ A = B. Theorem (The Principle of Functional Extensionality) f, g are functions : f = g ⇐⇒ dom(f) = dom(g) ∧ ( ∀x ∈ dom(f) : f(x) = g(x) ) ∀f ∀g ∀(a, b) : ((a, b) ∈ f ⇐⇒ (a, b) ∈ g) ⇐⇒ f = g. It may be that cod(f) ̸= cod(g). Hengfeng Wei (hfwei@nju.edu.cn) 1-10 Set Theory (III): Functions 2019 年 12 月 10 日 13 / 40
f:A→B g:C→D Q:lsf∩g a function? Theorem (Intersection of Functions) fng:(A∩C)→(BnD) Hengfeng Wei (hfweinju.edu.cn)1-10 Set Theory (III):Functions: 2019年12月10日14/40
f : A → B g : C → D Q : Is f ∩ g a function? Theorem (Intersection of Functions) f ∩ g : (A ∩ C) → (B ∩ D) Hengfeng Wei (hfwei@nju.edu.cn) 1-10 Set Theory (III): Functions 2019 年 12 月 10 日 14 / 40