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Set Theory: Surjective and Bijective

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i) If A is countable and f: A=>B is surjective, show that B is countable.
ii). Show that a function f: A=>B is bijective if, and only if, there is a function g: B=>A
with gf = 1_A and fg = 1_B.
iii) If f: A=>B, g: B=>C and h: C=>D are functions show that h(gf) = (hg)f.
iv) Let f: A=>B and g: B=>C be functions.
a) Show that if f is surjective and gf injective then g is injective.
b) Is it possible for gf to be injective when g is not? Explain.
v) Show that if A has n elements then P(A) has 2^n elements, for all n real numbers of N.

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We solve several problems in set theory. Surjective and bijectives are examined.

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(i) If A is countable, then we ...

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