Definition The set of all functions from X to Y: Yx={fIf:X→Y} y:y0={0} 0={0 X≠0：0x= 4口￥0,43，t夏，里Q0 Hengfeng Wei (hfweiinju.edu.cn)1-10 Set Theory (III):Functions 2019年12月10日9/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition The set of all functions from X to Y : Y X = {f | f : X → Y } ∀Y : Y ∅ = {∅} ∅ ∅ = {∅} ∀X ̸= ∅ : ∅ X = ∅ Hengfeng Wei (hfwei@nju.edu.cn) 1-10 Set Theory (III): Functions 2019 年 12 月 10 日 9 / 40
Definition The set of all functions from X to Y: Yx={fIf:X→Y} y:y0={0 00={0} X≠0：0X=0 4口￥0,43，t夏，里Q0 Hengeng Wei thkweionjn.edu.cn 1-10 Set Theory (II):Functions 2019 12 10 9/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition The set of all functions from X to Y : Y X = {f | f : X → Y } ∀Y : Y ∅ = {∅} ∅ ∅ = {∅} ∀X ̸= ∅ : ∅ X = ∅ Hengfeng Wei (hfwei@nju.edu.cn) 1-10 Set Theory (III): Functions 2019 年 12 月 10 日 9 / 40
Definition The set of all functions from X to Y: Yx={f|f:X→Y} 2={0,1}X≈P(X) 4口￥0,43，t夏里Q0 Hengong Wei Chkweinjn.ed.cn 1-10 Set Theory (III):Functions 2019 12 1010/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition The set of all functions from X to Y : Y X = {f | f : X → Y } 2 X = {0, 1} X ∼= P(X) Hengfeng Wei (hfwei@nju.edu.cn) 1-10 Set Theory (III): Functions 2019 年 12 月 10 日 10 / 40
Definition The set of all functions from X to Y: Yx={fIf:X→Y} 2X={0,1}X≈P(X) n- in out in OM比Ou七 4口，1①，43，t夏，30Q0 Hengeng Wei thkweionjn.ed.cn 1-10 Set Theory (III):Functions 2019 12 10 10/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition The set of all functions from X to Y : Y X = {f | f : X → Y } 2 X = {0, 1} X ∼= P(X) Hengfeng Wei (hfwei@nju.edu.cn) 1-10 Set Theory (III): Functions 2019 年 12 月 10 日 10 / 40
Definition The set of all functions from X to Y: Yx={f|f:X→Y} 4口￥0,43，t夏里0Q0 Hengfong Wei Chkweinjn.ed.cn 1-10 Set Theory (III):Functions 2019 12 10 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