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1. Do the following.
(1) Use the definition of Riemann Integral, prove that 01 xdx = 1/2.
(2) Let f (x) = 1 for rational numbers in [0,1]; f (x) = 0 for irrational numbers in [0,1].

Use the definition of Riemann Integral, show that f is not Riemann intergable in

[0,1].

2. Use mathematical induction to establish the well-order principle: Given a set S of

positive integers, let P(n) the propostion ”If n ∈ S, then S has a least element.”

3. Let f : X → Y be a mapping of nonempty space X onto Y . Show that f is 1-to-1 iff

thereisamappingg:Y →X suchthatg(f(x))=xforallx∈X.

4. Prove De Morgan’s law for arbitray unions and intersections.

5. Show that the set of all rational numbers is countable.

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