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
  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.
    • 4 years ago
    • 5