Let be a line in the plane given. Now are two sides of the line.
a) if any of the side is empty then will act as a boundary for . But as per definition of a plane, planes have no boundaries, it extends infinitely to all sides. Thus either of cannot be empty.
b) suppose one side of line have only finite number of points. Consider the point in that side which is the farthest point from the line . Now consider the line passing through parallel to line . Then there exist no point of in one side of the line , hence will be a bound for , which is a contradiction. Hence both contains infinite number of points.
1) Assume that (P. L. d) satisfies postulates 1-6 of neutral geometry. Let C P be...
Prove that (P;L; d) not satisfy postulate 6 of neutral geometry L = {1 c R313(a,b,c.), (u, v, w) є R3, such that I = {(a, b, cht.(u, v, w)|t є R)), and d: Px PR U, V, W T22 Postulate 6 (The Plane Separation Postulate). For any line l, the set of all points not on l is the union of two disjoint subsets called the sides ofl. If A and B are distinct points not on t, then...
3. Consider a geometry with five points a, b, c, d, and e. Let the lines consist of sets of two points. There are ten lines in this geometry. Show that, for every point P not on a line 1, there are at least two lines parallel to l.
(1) Assume the axioms of metric geometry. Let A, B, C, D be distinct collinear points. Let f : l → R be a coordinate function for the line l that crosses all of A, B, C, D. Suppose f(A) < f(B) < f(C) < f(D). Prove that AD = AB ∪ BC ∪ CD. (2) Assume the axioms of metric geometry. Let A, B, C, D be distinct collinear points. Suppose A ∗ B ∗ C and B ∗...
1. [1 points Let L S 10,1 and L E P. For strings x, y e 0,1 of the same length, let x田y denote the bitwise XOR of x and y-eg., 1000田0111 = 1111. Let ㈣ denote the length of z. Let L* L' = {x : 3y, y has lxl/2 ones and x89 E L). Show that L* E NP
Bayesian statistics question. Please do both parts. 3.6 Exponential family expectations: Let p(yo)-c(d)h(y) exptot(y)} be an a) Take derivatives with respect to ф of both sides of the equation b) Let p(d) x c(d)no enot0ф be the prior distribution for ф. Calculate exponential family model fp(jo) dy l to show that E[t(Y) d--d(φ)/c(d) dp(o)/ do and, using the fundamental theorem of calculus, discuss what must be true so that E-сф)/c(d)-to.
(Limit of functions) Let f : 2-» C be a function, and assume that D(a, r) C Q. We say that lim f(z) L Ď(a, 6) we have |f(z) Ll < e. if for any e > 0 there exists 6 > 0, such that for any z e (a) State the negation of the assertion "lim^-,a f(z) = L". (b) Show that lim- f(z) L if and only if for any sequence zn -» a, with zn a for...
Please show all work in READ-ABLE way. Thank you so much in advance. Problem 2.2 n and let X ε Rnxp be a full-rank matrix, and Assume p Note that H is a square n × n matrix. This problem is devoted to understanding the properties H Any matrix that satisfies conditions in (a) is an orthogonal projection matriz. In this problem, we will verify this directly for the H given in (1). Let V - Im(X). (b) Show that...
Let f : [a, b] → R and xo e (a,b). Assume that f is continuous on [a,b] \{x0} and lim x approaches too x0 f(x) = L (L is finite) exists. Show that f is Riemann integrable. 1. (20 pts) Let f : [a, b] R and to € (a,b). Assume that f is continuous on [a, b]\{ro} and limz-ro f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into...
2,Let X be a Poisson (mean-5) and Let Ybe a Poisson (mean-4). Let Z-X+Y.Find P(X-312-6) Assume X and Y are independent. 1 like to see answers for P(A), (B), P(AB), and and hence P(A B). Here A You can work out the probabilities (P(A).P(B),P(AB), and P(AIB) using your calculator, or Minitab or Mathematica. I dont need to see your commands.I just like to see the answers for the probabilities of ABABAIB You do item 1 lf your FSU id ends...
9 Geometry via calculus In this exercise you will see one way to use calculus to do grometry a) Here is one way to find the perpendicsler distance from a point to a line L (no caleulus yet) Let's say L has equation y-3r+2 and the point is (2.1) First, make a graph (picture) of the situation 2Now find an equation for the line AM through (2, 1) perpendicular to L (draw it first, of course). 3. Find the (coordinates...