Measurable functions

If f is measurable and almost everywhere nonzero, show that 1/f is measurable.

Measurable Functions

Show that a function f is measurable IF AND ONLY IF there exists a sequence (f_m) of ste functions such that f(x)=lim f_m(x) for almost all x. Please make sure to show the proof in both directions.

Problem set

Question 1 Consider the functions f(x) = x^2 and g(x) = square root of x, both with domain and co-domain R+, the set of positive real numbers. Are f and g inverse functions? Give a brief reason. Question 2 Given the Hamming distance function f: A X A -> Z defined on pairs of 8-bit strings, (where A is the set of 8-bit strings) select which of the following are correct (select zero or more). a. f is one-to-one b. f has range {0, 1, 2, 3, 4, 5, 6, 7, 8}. c. f is onto d. f is invertible. Question 3 Suppose set A = {0, 1, 3} and B = {2,4,6} in a universe U = {0,1,2,3,4,5,6,7}. Match the set names on the left with their membership lists... click for more

Pick all correct answers

Question Pick all correct answers: If f(x) = 3x^2 + 4ln(x) + 9, then f(x) is:... a. Big-Theta of g(x) = x^2 b. Big-Omega of g(x) = 1 c. An unimpressive time-cost function for a sorting algorithm d. Big-O of x^5 e. Big-Omega of x^5

Please select all the situations below that are POSSIBLE

Question Please select all the situations below that are POSSIBLE and do not mark those that are IMPOSSIBLE. Each list of numbers is a degree list (list of the degrees of all the vertices) of a graph. If there are extra restrictions - the graph is simple, or a tree, etc - it will be noted in the question. a. Graph, degrees: 1, 2, 3 b. Tree, degrees: 0, 0, 1, 1, 2, 3, 3. c. Tree, degrees: 1, 1, 1, 1, 2, 4 d. Simple graph, degrees: 1, 2, 3 e. Simple graph, degrees: 0, 0, 0, 2.