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9. Under what conditions could a geometry satisfy more than one of the parallel postulates? Explain. Could an incidence geometry satisfy more than one of the parallel postulates? Explain.

9. Under what conditions could a geometry satisfy more than one of the parallel postulates? Explain. Could an incidence geometry satisfy more than one of the parallel postulates? Explain.

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
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Question 9

clarified in this way, the kind of logical reasoning that characterizes mathematics becomes an incredibly powerful and effective tool. The history of mathematics is full of examples of surprising practical applications of mathematical ideas that were originally discovered and developed by people who were completely unaware of the eventual applications. EXERCISES 2.4 1. It is said that Hilbert once illustrated his contention that the undefined terms in a geometry should not have any inherent meaning by claiming that it should be possible to replace point by beer mug and line by table in the statements of the axioms. Consider three friends sitting around one table. Each person has one beer mug. At the moment all the beer mugs are resting on the table. Suppose we interpret point to mean beer mug, line to mean the table, and lie on to mean resting on. Is this a model for incidence geometry? Explain. Is this interpretation isomorphic to any of the examples in the text? 2. One-point geometry contains just one point and no line. Which incidence axioms does one-point geometry satisfy? Explain. Which parallel postulates does one-point geometry satisfy? Explain. 3. Consider a small mathematics department consisting of Professors Alexander, Bailey, Curtis, and Dudley with three committees: curriculum committee, personnel committee, and social committee. Interpret point to mean a member of the department, interpret line to be a departmental committee, and interpret lie on to mean that the faculty member is a member of the specified committee. (a) Suppose the committee memberships are as follows: Alexander, Bailey, and Curtis are on the curriculum committee; Alexander and Dudley are on the personnel 24 Chapter 2 Axiomatic Systems and Incidence Geometry committee; and Bailey and Curtis are on the social committee. Is this a model for Incidence Geometry? Explain. (b) Suppose the committee memberships are as follows: Alexander, Bailey and Curtis are on the curriculum committee; Alexander and Dudley are on the personnel committee; and Bailey and Dudley are on the social committee. Is this a model for incidence geometry? Explain. (c) Suppose the committee memberships are as follows: Alexander and Bailey are on the curriculum committee, Alexander and Curtis are on the personnel committee, and Dudley and Curtis are on the social committee. Is this a model for incidence geometry? Explain. 4. A three-point geometry is an incidence geometry that satisfies the following additional axiom: There exist exactly three points. (a) Find a model for three-point geometry. (b) How many lines does any model for three-point geometry contain? Explain. (c) Explain why any two models for three-point geometry must be isomorphic. (An axiomatic system with this property is said to be categorical.) 5. Interpret point to mean one of the four vertices of a square, line to mean one of the sides of the square, and lie on to mean that the vertex is an endpoint of the side. Which incidence axioms hold in this interpretation? Which parallel postulates hold in this interpretation? 6. Draw a schematic diagram of five-point geometry (see Example 2.2.5). 7. Which parallel postulate does Fano's geometry satisfy? Explain. 8. Which parallel postulate does the three-point line satisfy? Explain. 9. Under what conditions could a geometry satisfy more than one of the parallel postulates? Explain. Could an incidence geometry satisfy more than one of the parallel postulates? Explain. 10. Consider a finite model for incidence geometry that satisfies the following additional axiom: Every line has exactly three points lying on it. What is the minimum number of points in such a geometry? Explain your reasoning. 11. Find a finite model for Incidence Geometry in which there is one line that has exactly three points lying on it and there are other lines that have exactly two points lying on them. 12. Find interpretations for the words point, line, and lie on that satisfy the following conditions. (a) Incidence Axioms 1 and 2 hold, but Incidence Axiom 3 does not. (b) Incidence Axioms 2 and 3 hold, but Incidence Axiom 1 does not. (c) Incidence Axioms 1 and 3 hold, but Incidence Axiom 2 does not. 13. For any interpretation of incidence geometry there is a dual interpretation. For each point in the original interpretation there is a line in the dual and for each line in the original there is point in the dual. A point and line in the dual are considered to be incident if the corresponding line and point are incident in the original interpretation. (a) What is the dual of the three-point plane? Is it a model for incidence geometry? (b) What is the dual of the three-point line? Is it a model for incidence geometry? (c) What is the dual of four-point geometry? Is it a model for incidence geometry? (d) What is the dual of Fano's geometry? 2.5 THEOREMS, PROOFS, AND LOGIC We now take a more careful look at the third part of an axiomatic system: the theorems and proofs. Both theorems and proofs require extra care. Most of us have enough experience with mathematics to know that the ability to write good proofs is a skill that must be learned, but we often overlook the fact that a necessary prerequisite to good proof writing is good statements of theorems. A major goal of this course is to teach the art of writing proofs and it is not expected that the reader is already proficient at it. The main way in which one learns to write proofs Section 2.5 Theorems, proofs, and logic 25 is by actually writing them, so the remainder of the book will provide lots of opportunities - -- -.1 -.---1.-1---- ---- 1 11. --- 41. --- -:--1--
Transcribed Image Text:clarified in this way, the kind of logical reasoning that characterizes mathematics becomes an incredibly powerful and effective tool. The history of mathematics is full of examples of surprising practical applications of mathematical ideas that were originally discovered and developed by people who were completely unaware of the eventual applications. EXERCISES 2.4 1. It is said that Hilbert once illustrated his contention that the undefined terms in a geometry should not have any inherent meaning by claiming that it should be possible to replace point by beer mug and line by table in the statements of the axioms. Consider three friends sitting around one table. Each person has one beer mug. At the moment all the beer mugs are resting on the table. Suppose we interpret point to mean beer mug, line to mean the table, and lie on to mean resting on. Is this a model for incidence geometry? Explain. Is this interpretation isomorphic to any of the examples in the text? 2. One-point geometry contains just one point and no line. Which incidence axioms does one-point geometry satisfy? Explain. Which parallel postulates does one-point geometry satisfy? Explain. 3. Consider a small mathematics department consisting of Professors Alexander, Bailey, Curtis, and Dudley with three committees: curriculum committee, personnel committee, and social committee. Interpret point to mean a member of the department, interpret line to be a departmental committee, and interpret lie on to mean that the faculty member is a member of the specified committee. (a) Suppose the committee memberships are as follows: Alexander, Bailey, and Curtis are on the curriculum committee; Alexander and Dudley are on the personnel 24 Chapter 2 Axiomatic Systems and Incidence Geometry committee; and Bailey and Curtis are on the social committee. Is this a model for Incidence Geometry? Explain. (b) Suppose the committee memberships are as follows: Alexander, Bailey and Curtis are on the curriculum committee; Alexander and Dudley are on the personnel committee; and Bailey and Dudley are on the social committee. Is this a model for incidence geometry? Explain. (c) Suppose the committee memberships are as follows: Alexander and Bailey are on the curriculum committee, Alexander and Curtis are on the personnel committee, and Dudley and Curtis are on the social committee. Is this a model for incidence geometry? Explain. 4. A three-point geometry is an incidence geometry that satisfies the following additional axiom: There exist exactly three points. (a) Find a model for three-point geometry. (b) How many lines does any model for three-point geometry contain? Explain. (c) Explain why any two models for three-point geometry must be isomorphic. (An axiomatic system with this property is said to be categorical.) 5. Interpret point to mean one of the four vertices of a square, line to mean one of the sides of the square, and lie on to mean that the vertex is an endpoint of the side. Which incidence axioms hold in this interpretation? Which parallel postulates hold in this interpretation? 6. Draw a schematic diagram of five-point geometry (see Example 2.2.5). 7. Which parallel postulate does Fano's geometry satisfy? Explain. 8. Which parallel postulate does the three-point line satisfy? Explain. 9. Under what conditions could a geometry satisfy more than one of the parallel postulates? Explain. Could an incidence geometry satisfy more than one of the parallel postulates? Explain. 10. Consider a finite model for incidence geometry that satisfies the following additional axiom: Every line has exactly three points lying on it. What is the minimum number of points in such a geometry? Explain your reasoning. 11. Find a finite model for Incidence Geometry in which there is one line that has exactly three points lying on it and there are other lines that have exactly two points lying on them. 12. Find interpretations for the words point, line, and lie on that satisfy the following conditions. (a) Incidence Axioms 1 and 2 hold, but Incidence Axiom 3 does not. (b) Incidence Axioms 2 and 3 hold, but Incidence Axiom 1 does not. (c) Incidence Axioms 1 and 3 hold, but Incidence Axiom 2 does not. 13. For any interpretation of incidence geometry there is a dual interpretation. For each point in the original interpretation there is a line in the dual and for each line in the original there is point in the dual. A point and line in the dual are considered to be incident if the corresponding line and point are incident in the original interpretation. (a) What is the dual of the three-point plane? Is it a model for incidence geometry? (b) What is the dual of the three-point line? Is it a model for incidence geometry? (c) What is the dual of four-point geometry? Is it a model for incidence geometry? (d) What is the dual of Fano's geometry? 2.5 THEOREMS, PROOFS, AND LOGIC We now take a more careful look at the third part of an axiomatic system: the theorems and proofs. Both theorems and proofs require extra care. Most of us have enough experience with mathematics to know that the ability to write good proofs is a skill that must be learned, but we often overlook the fact that a necessary prerequisite to good proof writing is good statements of theorems. A major goal of this course is to teach the art of writing proofs and it is not expected that the reader is already proficient at it. The main way in which one learns to write proofs Section 2.5 Theorems, proofs, and logic 25 is by actually writing them, so the remainder of the book will provide lots of opportunities - -- -.1 -.---1.-1---- ---- 1 11. --- 41. --- -:--1--
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