Math Review
Homework #1
1- Let Z = {..., -3, -2, -1, 0, 1, 2, 3,...}denote the set of integers;
Q = {m/n: mZ, nZ, n  0} denote the set of rational numbers.
Show that the equation x2 = 2 is not satisfied for any rational xQ. (Hint: Use proof by contradiction).
2-
. Suppose that {Si}, iI, is a family of subsets of some set T. Prove that
iI(T \ Si) = T \ (iI Si).
. Suppose that S1 and S2 are convex sets in Rn. Let (1 S1 + 2 S2) = {xRn: x = 1 x1 + 2 x2,
x1S1, x2S2}, where iR, i = 1, 2. Show that (1 S1 + 2 S2) is a convex set.
3- Let A be the set of even integers, B the set of odd integers, C the set of integers from 1 to 10, and D the
set of nonnegative real numbers. Describe CA, CB, C\B, AD, BD, AB, and AB.
4- Consider the following functions f: XY, where XR, YR,
a) f(x) = 3x – 7, X=Y=R; b) f(x) = x2 – 1, X=R, y-1;
c) f(x) = ex, X=R, y>0;
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d) f(x) = x – x, X=Y=R; e) f(x) = x/(x + 1), X=R, -.5y.5; f) f(x) = x3, X=Y=R;
g) f(x) = 1/x, X=Y=R\{0}; h) f(x) = (x -1).5, x1, y0;
i) f(x) = x e-x, X=R, y<(1/e);
For each of the above functions,
. Which function is one-to-one?
. For the functions that are one-to-one, write the expression for the inverse.
5- Find the supremum and the infimum (if they exist) of each of the following sets of real numbers:
a) the natural numbers N = {1, 2, 3, ...}; b) {1/1, 1/2, 1/3, ..., 1/n, ...};
c) {1/2, -1/2, 2/3, -2/3, 3/4, ...};
d) {xR: 0 < x < 1};
e) {xR: 0  x < 1};
f) {xR: 0  x  1}.
6- Let A be a nonempty set of real numbers. Define -A = {-x: xA}. Prove that:
inf A = - sup(-A) if A is bounded from below;
sup A = - inf(-A) if A is bounded from above.
7- Let nN, where N = {1, 2, 3, ...} is the set of natural numbers. Prove that the sum of the first n natural
numbers equals n(n+1)/2:
1 + 2 + 3 + ... + n = n(n+1)/2.
(Hint: use proof by induction).
8- Let nN, n  1, where N is the set of natural numbers. Given that aR\{1},
. Prove that:
S(n)  1 + a + a2 + a3 + ... + an = [1-an+1]/(1-a).
. Evaluate limnS(n).