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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].
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.”
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.
Prove De Morgan’s law for arbitray unions and intersections.
Show that the set of all rational numbers is countable.
