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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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