M CO diTng hay khong tren w la yeu to quyet dinh su ton tai thuat
90 M^y Turing vh Thuat toan
I them chung vao danh sach, ta se lap lai y ^ |. Ta trfinh lam the
bSing each bo qua phan tvt k h i no gay nen su lap laị N h u vay ta chi
them vao danh sach hai phan t i i mdi f va |. Tiep tuc each lam nay t a t h u dudc danh sach tat ca cac phan t i i cua Q + , nhiing so dugc
khoanh tron trong Hinh 1.8. . Q
K h i chiing k i l n sii tUOng ling giQa N va Q+ t a chot nghi r^ng hai tap v6 han b i t ky c6 thg diiOc chiing to la c6 cung liic luong, b^ng each b i i u t h i cac phep tuong ilng giOa tiing tap ay v6i tap N . Tuy nhien, t o n t a i nhieu tap v6 han ma khQng tUOng ling duoc v6i tap N . Cac tap nay that to Idn. Nhiing tap nhil vay dUdc goi la
khong dim dtidc.
T^p cac so thiic la mot t h i du ve tap khong dem duoc. So thixc
la so ma bieu dien diioc diidi dang thap phan. M o i so hiiu t i deu la so thiic. Cac so TT = 3,1415926... va = 1,4142135... cung la cac • so thiic. • .
Gia sii R la tap tat ca cac so thiic. Cantor da chiing m i n h r^ng
tap R la khong dem diiOc b^ng phuong phdp dudng cheo do chinh
ong de xuat.
Dinh ly 1.4.4 R la khong dim duac.
CMng minh Dg chiing to R la khong dem diioc, t a phai chiing
minh r^ng khSng t o n tai phep tiiong ilng giita N va R. Dieu nay duoc chiing m i n h b^ng phan chiing. Gia siJt ton t a i phep tUOng ilng / giiia N va R. Nhiem v u c u a ta la chiing to rSing / khong lam tron nhiing gi n6 phai thuc hien. Dg t r d thanh mot phep tuong ilng, / phai ghep doi dudc moi thanh phan cua N vdi moi thanh phan ciia
R. Nhung t a se chi ra phan t i l x trong R ma khong duqc ghep doi
vdi bat cil phan t i l nao cua R. Do la dieu mau thuan ma t a c^n den.
Vigc chi ra phan t i l x duqc thiic hien b^ng each kien thiet thuc sụ Ta chpn tiing chC so dg tao nen x sao cho x khac v6i moi s6 1
thuc da duoc ghep doi v6i phan t i l cua N .
1.4 Mot vdi bai todn khong gidi duac 91
Ta CO thg m i n h hoa y tudng nay b^ng each cho t h i dụ Gia sii ton t a i phep tuong ling /. Cho / ( I ) = 1, /(2) = 3,14159..., /(3) = 55,55555 ..., va cil nhu the ta c6 thgm cac gia t r i cua /. K h i d6 / ghep doi so 1 vdi 1, s6 2 vdi 3,14159..., s6 3 vdi 55,55555...,
va cii tiep tuc nhu vaỵ Bang sau day the hien nhitng gia t r i dau tign ciia phep tuong ling gia dinh / giiia N va R.
n f{n)
1 1 0 0 0 0 0 . . . 2 3 1 4 1 5 9 . . . 2 3 1 4 1 5 9 . . . 3 55 5 5 5 5 5 . . . 4 0 1 2 3 4 5 . . .
Bay gid, tren cd sci nhiing gia t r i trong bang nay, ta xay dung
so X bang each bigu dign no dudi dang thap phan bdi cac chii so
ttr 0 dgn 9, va theo do tat ca cac chfl so quan trong n&m 5 phan
thap phan. Muc tigu ciia ta la bao dam r^ng x ^ /(n) doi vdi moi n. De X 7^ / ( I ) , thanh phan thap phan t h i i nhat cua X cd thg
duoc gan bat cil chU so nao m i i n sao khac 0, day la thanh phan thap phan dau tien cua / ( I ) = 1,00000.... M o t each tiiy hiing, ta chon chu so 1 vao vi t r i thap phan dau tign cua x. Dg x ^ /(2),
thanh phan thap phan t h i l hai cua x cd thg dugc gan bat cil chii
s6 nao mign sao khac 4, day la thanh phan thap phan t h i i hai ciia
/(2) = 3,14159... . Mot each tiiy hiing, ta chon chii so 2 vao v i t r i
thap phan t h i l hai cua x. Tiep tuc each nay theo dudng cheo cua
bang gia t r i cua /, bat dau tif chii so thap phan t h i l nhat cua / ( I )
cheo xu6ng chii so thap phan t h i i n cua /(n) vdi n = 2,3,..., ta cd dugc tat ca cac phanh phan thap phan cua x, n h u trong bang sau day: