Prove de morgans law for arbitray unions and intersections

Question

Assignment

1. Do the following.

(1) Use the definition of Riemann Integral, prove that ₀∫¹ 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 such that g(f(x))=x for all x ∈ X.

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

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