let f be a mapping from e into f. show that
1. f?1( ) = ,
2. f?1(f) = e,
3. f?1(b \ c) = f?1(b) \ f?1(c),
4. f?1(
s
i2i bi) =
s
i2i f?1(bi),
5. f?1(
t
i2i bi) =
t
i2i f?1(bi),
for all subsets b,c,bi of f.
2.2 show that x 7! e?x is a bijection from r+ onto (0, 1]. show that x 7!
log x is a bijection from (0,1) onto r. (incidentally, log x is the logarithm
of x to the base e, which is nowadays called the natural logarithm.
we call it the logarithm. let others call their logarithms “unnatural.”)
2.3 let f be defined by the arrows below:
1 2 3 4 5 6 7 · · ·
# # # # # # #
0 ?1 1 ?2 2 ?3 3 · · ·
this defines a bijection from n onto z. using this, construct a bijection
from z onto n.
2.4 let f : n×n 7! n be defined by the table below where f(i, j) is the entry
in the ith row and the jth column. use this and the preceding exercise to
construct a bijection from z × z onto n.
6 sets and functions
. . . j 1 2 3 4 5 6 · · ·
i
. . .
1 1 3 6 10 15 21
2 2 5 9 14 20
3 4 8 13 19
4 7 12 18
5 11 17
6 16
...
2.5 functional inverses. let f be a bijection from e onto f. then, for each
y in f there is a unique x in e such that f(x) = y. in other words, in
the notation of (2.1), f?1({y}) = {x} for each y in f and some unique
x in e. in this case, we droping some brackets and write f?1(y) = x. the
resulting function f?1 is a bijection from f onto e it is called the functional
inverse of f. this particular usage should not be confused with the
general notation of f?1. (note that (2.1) defines a function f?1 form f
into e, where f is the collection of all subsets of f and e is the collection
of all subsets of e.)
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
الرجوع الى لوحة التحكم
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