Show that in general the sets X X ( YxZ) and
(X X Y)XZ are different but that there is
a natural Correspondence between each of them
and X X Y X Z
be mapping of a nonempty space
X into Y. show that fis one-to-one if and only
if there is a mapping g:Y
gof is the identity mapon X, that is, such that
9(f(x) =x for all XE X
(a ishow that if f maps X into Y and ACX
and B C Y, then [FT[B1] CB
Give examples to show that we need not have
show that if fmaps X onto Y and BCY,
then F[F([B]] = B
X Y be a mapping onto Y.
Then thone is a mapping g: Y X such that
fog is the identity map on Y. [Apply the axiom
of choice to the Collection (A:(Zyey) with
prove the proposition which is [the set
of all rational numbers is Countable] by using
the following /propositions S:
I O|Every subset of a Countable Set is Countable].
[letA be acountable set. Then the set of all
finite sequences from A is also Countable ]
Hint: the mapping P/q
is 9 function whose range is the set of rational
numbers and whose domain is a subset
of the set of finite sequence from IN. .
let X be an Abekian group undert. Then
IIE is Compatible with + iff x=x' implies
x+ y ill x'ty. The induced operation then makes
the quotient Space into a group.
Let Y be the Set of ordinals less than the
first Un Countable ordinal; i.e Y= = (xEX : < 52}
Show that every Countable subset Eof Y has an upper
bound in Yand hence a least upper
b is an upper bound for E if xIt is a least upper bound if b< b* for each upper
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