Mathematics 713 Assignment #2

    1.  Let X be a finite set and D any collection of subsets of X.
        Prove or find counter example to the claim that D must be
        a monotone class.

    2.  For two sets A and B show that the following statements
        are equivalent

        a.  A ⊆ B
        b.  A ∪ B = B
        c.  A ∩ B = A

        in otherwords, show A being a subset of B is equivalent
        to the union of A with B being equal B and that this is
        equivalent to the intersection of A with B equal to A.

    3.  Let A be an uncountable set and B be a countable subset
        of A.  Show that A is set equivalent to A\B.

    4.  [Extra Credit]  Let sin(x) be the sine of x where x is
        expressed in radians.  Let E be the set of cluster points
        of the sequence An = sin(n) for n=1,2,3,....  Prove or
        disprove the claim that E is equal to the interval [-1,1].

    5.  Work problem 2.40 in McDonald and Weiss.

    6.  Suppose f:[0,1]->R is continuous and c satisfies 0 < c < 1.

        a.  If f(c) > 0 show that there is h > 0 such that f(x) > 0 
            for every x with |x-c| < h.
        
        b.  If
                 1       2
                ∫  (f(x))  dx = 0
                0

            show that f(x) = 0.  In otherwords, if the integral of 
            the square of f is zero then f must be zero.