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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 let f: X Y 2 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 X such that gof is the identity mapon X, that is, such that 9(f(x) =x for all XE X 3 (a ishow that if f maps X into Y and ACX -1 and B C Y, then [FT[B1] CB and {[[fA]]DA . b Give examples to show that we need not have equality C show that if fmaps X onto Y and BCY, / then F[F([B]] = B 4 Let f: 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 = 5 prove the proposition which is [the set of all rational numbers is Countable] by using two the following /propositions S: I O|Every subset of a Countable Set is Countable]. 2 [letA be acountable set. Then the set of all finite sequences from A is also Countable ] Hint: the mapping P/q P1q <1,1,37 O 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. . 2 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. 7 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 bound. (An element b is an upper bound for E if xIt is a least upper bound if b< b* for each upper bound b.). 3

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