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Synthetic Geometry

Synthetic geometry differs from other branches of geometry because it focuses on pure geometrical contents and draws conclusions through the use of axioms, logical arguments and theorems. Synthetic geometry analyzes the incidence relationship between objects such as lines and planes. These objects and their relationship of incidence are known as the primitives of synthetic geometry.

Axiomatic geometry is sometimes used interchangeably with the term synthetic geometry. In mathematics, the concept of an axiom is critical to the study of Euclidean geometry. Axioms are statements which are used to describe and prove the primitives of synthetic geometry. Essentially, these are just basic statements which have not yet been accepted or proven. For example, the statement “for any two distinct points, there is only one line which passes through both of these points”, is an axiom.

In conjunction with axioms, theorems are used in synthetic geometry. Theorems differ from axioms because they represent proven statements. Theorems can be proven by axioms which are thought to be accepted or by other theorems which exist. In order to formulate conclusions regarding mathematical problems and conduct geometric computations, theorems are utilized in synthetic geometry. Furthermore, logical arguments are used as a mode of proof through persuasion.

Evidently, synthetic geometry is one of the many distinct branches of geometry, just like analytic and algebraic geometry, which is used to understand geometrical structures. Thus, understanding the proper use of axioms, logical arguments and theorems is essential to this field of thought. 

Uses of the Pythagorean Theorem

Geometry is a very broad field of mathematics composed of a wide range of tools that can be used for problem solving. In this module, you are going to research three examples of the implementation of geometry that would employ the use of the Pythagorean Theorem as a problem-solving tool. The examples you find can come from s

continuous problems

8. Use the definition of f(x) ® ¥ as x ® ¥ to show that f(x) = x + 2 ® ¥ as x ® ¥ 9. (a) Show that f(x) = tends to ¥ as x ® -2+ . (b) Show that f(x) = tends to ¥ as x ® 4- 10. Let f(x) = . Show that f is not continuous at x = 5.

Give a counterexample.

1. (a) Prove that if = ¥ then = 0 (b) Give a counterexample to show that the converse (if = 0 then = ¥) is false. 2. Give an example of a sequence {an} satisfying all of the following: {an} is monotonic 0 < an < 1 for all n and no two terms are equal =

Mathematical Reasoning: Example Problem

The Pythagorean Theorem for right triangles states that if a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then a2 + b2 = c2. Prove that if m is a real number and m, m + 1, and m + 2 are the lengths of the three sides of a right triangle, then m = 3.

Symbolic Form

1. U= {1,2,3,4,5,6,7,8 } A= {1,2,4,5,8 } B= {2,3,4,6 } Determine the following and explain your answer A ∩B 2. U= {3,5,7,9,11,13,15} A= {3,5,7,9} B= {7,9,11,13} C= {3,11,15} Determine the following and explain your answer A ∩B 3. Write the statem

set builder notation concepts

1. Are the two sets below equal, equivalent, neither or both? Explain your answer. V = {p, q, r, s}; W = {boy, dad, father, kid} 2. Given U = {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, A = {15, 17, 21, 23, 25}, and B = {16, 17, 18, 22, 25}. Find A′ ∩ B′. Show your work, finding A', then B' then find their intersec

Truth Table Statement and Argument

Write the statements in symbolic form p: The temperature is 90deg. q: The air conditioner is working r: The apartment is hot. In exercises 7-16 construct a truth table for the statements: ~ (~p←>q) Sec. 3.5 #40 Translate the argument into symbolic form and (b) determine if the argument is valid or invalid. You may

Protective Planes Spaces

4. a) Show that the projective space with a disk removed (more precisely, the image of a disk on S2 under the natural projection pi : S2 -> P2 removed) is a M¨obius band. b) Let M be the connected sum of two copies of the projective space formed by removing a disk from each copy of P2 and gluing the resulting surfaces together

Compact metric space and contraction maps

Here is the question. Let (X,d) be a compact metric space, and let Con(X) denote the set of contraction maps on X. We shall define the distance between two maps f,g which belongs to Con(X) as follows : d_c (f,g) = sup d ( f(x) , g(x) ) for any x that belongs to X. a) Show that d(y_f,y_g) <= d_c(f,g)/(1-min(cf,cg

Taylor's Theorem

Please see the attached file for the fully formatted problems. 1 Write down the Taylor polynomial of , . Use Taylor's theorem to show that = 2. Given that , 1 < e < 3. Do the followings. a) Write down the Taylor polynomial of and show that for and b) Use a) to show that for al

Nullhomotopic Mappings and Contractible Spaces

I am having problems proving this fact. A space X is contractible if and only if every map f:X to Y (Y is arbitrary) is nullhomotopic. Similarly show X is contractible iff ever map f:Y to X is nullhomotopic. In the first case if Y=X then see that the identity map on X is nullhomotopic. But Im not sure how to proceed for th

Path connected problems

(See attached file for full problem description with proper symbols) --- ? Show that, for , the sphere is path connected. ? Show that if f:X->Y is a continuous map between topological spaces and X is path connected, then the image f(Y) is also path connected. ---

Subset Game involving Intervals and Subintervals

In the following infinite game, Alice and John take turns moving. First, Alice picks a closed interval I<sub>1</sub> of length <1. Then, Bob picks a closed subinterval I<sub>2</sub> which is a subset of or equal to I<sub>1</sub>, of length 1/2. Next, Alice picks a closed subinterval I<sub>3</sub> which is a subset of or equal t

Vertices in Regular Solids

For each of the 5 regular polyhedra, enumerate the number of vertices (v), edges (e), and faces (f), and then evaluate the quantity v - e + f. (One of the most interesting theorems relating to any convex polyhedron is that v - e + f = 2. -> This was given as part of the problem. I am not sure if it is of any value when solving t

Identifying types of conics

For each of the equations identify the type of conic it is and list the major features associated with this type of conic Major Features Circles - Find the coordinates of the center - Find the length of the radius - Find the length of the diameter Parabolas - Find the coordinates of the vertex - Find the coordinates

Chart movement using latitude and longitude

To chart the movement of a polar bear, scientists attached a radio transmitter to its neck. Two tracking stations monitor the radio signals. Station B is 10 miles directly east of station A. On Monday, station A measured the direction of the bear at N43degreesE, and, station B, at N30degreesW. Three days later, the directions to