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Geometry and Topology

Topology and mapping functions

Topology Sets and Functions (XXXIV) Functions Let X be a non-empty set. The identity mapping ix on X is the mapping of X onto itself defined by ix(x) = x

Topology and functions

Topology Sets and Functions (XXXIII) Functions Two mappings f : X --> [Y and g : X --> Y are said to be equal ( and we write this f = g ) if f(x) = g(x) for every x in X.

The First Second Mapping

Topology Sets and Functions (XXX) Functions Consider an arbitrary mapping f : X -->Y. Prove the main property of the second set mapping:

Proving the Main Property of a Second Set Mapping

Consider an arbitrary mapping f : X → Y. Suppose that f is a one-to-one onto. Prove the main property of the second set mapping: B1 is a subset of B2 implies f^(-1)(B1) is a subset of f^(-1)(B2). See the attached file for format

Main property of the first Set Mapping

Consider an arbitrary mapping f : X -> Y. Prove the main property of the first set mapping: f(intersection_i A_i) = intersection_i f(A_i) Please see attachment if the symbols are compromised.

Vector Analysis: Parallelepiped

(a) Show that (A x B) dot C is equal to the volume of the parallelepiped shown. (b) Without doing extensive calculations show that (A x B) dot C = A dot (B x C) = B dot (C x A). Please view the attachment to see the full question and proper formatting.

Perimeter of a Rectangle

3. A rectangle is a parallelogram with four right angles. A rectangle has a width of 15 feet and a diagonal of a length 22 feet; how long is the rectangle? What is the perimeter of the rectangle? Round to the nearest foot. See attached file for full problem description.

Main property of the first set mapping

Consider an arbitrary mapping f : X --> Y. Prove the main property of the first set mapping: A_1 is a subset of A_2 implies that f(A_1) is a subset of f(A_2). The attached file contains the symbol version of the above statement for clarity.

Prove a First Set Mapping

Consider an arbitrary mapping f : X -->Y. Prove the main property of the first set mapping: f(X) is a subset of Y. See attachment for fully-formatted version of the question, should your display not include the symbols.

Maximum Volume of an Open-Top Box

A square sheet of cardboard 24 inches on a side is made into a box by cutting squares of equal size from each corner of the sheet and folding the projecting flaps into an open top box. What should be the length of the edge of any of the cutout squares to give the box maximum volume? 4 inches 4.5 inches

Strain Tensors and Elongation

Please see the attached file for the fully formatted problems. The components of the strain tensor are: (5 3 0 3 4 -1 ....... x 10^-4 0 -1 2) What is the elongation in the (2, 2, 1) direction? What is the change of angle between two perpendicular lines (in the undeformed state) in the directions (2, 2, 1) a

Basic Graphing

46. Technology. Driving down a mountain, Tom finds that he has descended 1800 ft in elevation by the time he is 3.25 mi horizontally away from the top of the mountain. Find the slope of his descent to the nearest hundredth. Section 7.2 pp. 626-627 16,20,28 16. Find the slope of any line perpendic

Area, Perimeter and Polar Equation of an Ellipse

Consider the formed by the parametric equations: x=5cosθ+3; y=4sinθ a) estimate the perimeter of the ellipse. b) estimate the area enclosed by the ellipse. c) find a polar equation for the ellipse.

Real Life Applications of Geometry

You are part of a panel of parents, teachers, and administrators working to revise the geometry curriculum for the local high school. On tonight's agenda, you will be brainstorming creative ways to teach surface area and volume. The teachers are especially interested in methods which will help the students connect geometry to li

Application of laws of cosine and sin rule

At 1:00pm Jill left home traveling 45mph on a bearing of N 40 degrees W. At 1:30pm John left traveling 50mph on a bearing of S 75 degrees W. A) Illustrate the positions of Jill and John at 3:00pm. (I calculated that Jill would be 90 miles away and John would be 75 miles away) B) Find the measure of the angle between their

Subspace - Linear Equation Questions

Linear Equation Questions. See attached file for full problem description. Let S be the subspace.... Explain why this linear equation represents a subspace and find a basis for it. Clearly explain why this subspace is a plane. Find two orthogonal vectors in the plane. Make the set you found orthonormal. Explain why your

Find the set of points of convergence of a given filter on an infinite set X with the cofinite topology. Prove that a space is compact if and only if every open cover has an irreducible subcover.

1. Let X be an infinite set, let T be the cofinite topology on X, and let F be the filter generated by the filter base consisting of all the cofinite subsets of X. To which points of X does F converge? 2. Let X be a space. A cover of X is called irreducible if it has no proper subcover. (a) Prove that X is compact if and o

Sets and Sequences

2.) If " S " is the set of all "x" such that 0≤x≤1, what points, if any, are points of accumulation of both "S" and C(S)? 3.) Prove that any finite set is closed. 5.) Prove that, if "S" is open, each of its points is a point of accumulation of "S". 1.) Suppose "S" is a set having the number "M" as its least up

Volume and Surface Area

For problem A, find the area of the triangle. For problem B, find the volume of the cylinder and volume of cylinder if applicable. See attached file for full problem description.