Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
2. Relationship between neutral geometry and incidence geometry. (a) Explain how the axioms stated in this chapter imply that there exists at least one line. (b) Explain how the existence of three noncollinear points (Incidence Axiom I-3) can be deduced from the axioms stated in this chapter. (c) Explain why every model for neutral geometry must have infinitely many points. (d) Explain why every model for neutral geometry must have infinitely many lines. (e) Explain why every model for neutral geometry must also be a model for incidence geometry. (f) Explain why all the theorems of incidence geometry must also be theorems in neutral geometry.

2. Relationship between neutral geometry and incidence geometry. (a) Explain how the axioms stated in this chapter imply that there exists at least one line. (b) Explain how the existence of three noncollinear points (Incidence Axiom I-3) can be deduced from the axioms stated in this chapter. (c) Explain why every model for neutral geometry must have infinitely many points. (d) Explain why every model for neutral geometry must have infinitely many lines. (e) Explain why every model for neutral geometry must also be a model for incidence geometry. (f) Explain why all the theorems of incidence geometry must also be theorems in neutral geometry.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Question 2

proved that the axioms of neutral geometry are consistent. For now we will proceed in the faith that there is a model and will revisit the question of consistency in a later chapter. Even though we will treat the three geometries axiomatically, we will still want to draw diagrams that illustrate the relationships being discussed. A diagram should be viewed as one possible interpretation of a theorem. Both the theorem statements and the proofs should stand alone and should not depend on the diagrams. The purpose of a diagram is to help us to visually organize the information being discussed. Diagrams add tremendously to our intuitive understanding, but they are only meant to illustrate, not to narrow the range of possible interpretations. EXERCISES 3.7 1. Check that the trivial geometry containing just one point and no lines satisfies all the postulates of neutral geometry except the Existence Postulate. Which parallel postulate 68 Chapter 3 Axioms for Plane Geometry is satisfied by this geometry? (This exercise is supposed to convince you of the need for the Existence Postulate.) 2. Relationship between neutral geometry and incidence geometry. (a) Explain how the axioms stated in this chapter imply that there exists at least one line. (b) Explain how the existence of three noncollinear points (Incidence Axiom I-3) can be deduced from the axioms stated in this chapter. (c) Explain why every model for neutral geometry must have infinitely many points. (d) Explain why every model for neutral geometry must have infinitely many lines. (e) Explain why every model for neutral geometry must also be a model for incidence geometry. (f) Explain why all the theorems of incidence geometry must also be theorems in neutral geometry. СНАРТER 4 Neutral Geometry 4.1 THE EXTERIOR ANGLE THEOREM AND EXISTENCE OF PERPENDICULARS 4.2 TRIANGLE CONGRUENCE CONDITIONS 4.3 THREE INEQUALITIES FOR TRIANGLES 4.4 THE ALTERNATE INTERIOR ANGLES THEOREM
Transcribed Image Text:proved that the axioms of neutral geometry are consistent. For now we will proceed in the faith that there is a model and will revisit the question of consistency in a later chapter. Even though we will treat the three geometries axiomatically, we will still want to draw diagrams that illustrate the relationships being discussed. A diagram should be viewed as one possible interpretation of a theorem. Both the theorem statements and the proofs should stand alone and should not depend on the diagrams. The purpose of a diagram is to help us to visually organize the information being discussed. Diagrams add tremendously to our intuitive understanding, but they are only meant to illustrate, not to narrow the range of possible interpretations. EXERCISES 3.7 1. Check that the trivial geometry containing just one point and no lines satisfies all the postulates of neutral geometry except the Existence Postulate. Which parallel postulate 68 Chapter 3 Axioms for Plane Geometry is satisfied by this geometry? (This exercise is supposed to convince you of the need for the Existence Postulate.) 2. Relationship between neutral geometry and incidence geometry. (a) Explain how the axioms stated in this chapter imply that there exists at least one line. (b) Explain how the existence of three noncollinear points (Incidence Axiom I-3) can be deduced from the axioms stated in this chapter. (c) Explain why every model for neutral geometry must have infinitely many points. (d) Explain why every model for neutral geometry must have infinitely many lines. (e) Explain why every model for neutral geometry must also be a model for incidence geometry. (f) Explain why all the theorems of incidence geometry must also be theorems in neutral geometry. СНАРТER 4 Neutral Geometry 4.1 THE EXTERIOR ANGLE THEOREM AND EXISTENCE OF PERPENDICULARS 4.2 TRIANGLE CONGRUENCE CONDITIONS 4.3 THREE INEQUALITIES FOR TRIANGLES 4.4 THE ALTERNATE INTERIOR ANGLES THEOREM
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS