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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.

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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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2.4 question 12

uare, pomt 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? HEOREMS, 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 for practice. This section simply lays out a few basic principles and then those principles will be put to work in the rest of the course. The brief introduction provided in this section will not make you an instant expert at writing proofs, but it will equip you with the basic tools you need to get started. You should refer back to this section as necessary in the remainder of the course. Mathematical language An essential step on the way to the proof of a theorem is a careful statement of the theorem in clear, precise, and unambiguous language. To illustrate this point, consider the following proposition in incidence geometry. Proposition 2.5.1. Lines that are not parallel intersect in one point. Compare that statement with the following. Restatement. If l and m are two distinct lines that are not parallel, then there exists exactly one point P such that P lies on both l and m. Both are correct statements of the same theorem. But the second statement is much better, at least as far as we are concerned, because it clearly states where the proof should begin (with two lines l and m such that l + m and l f m) and where it should end (with a point P that lies on both l and m). This provides the framework within which we can
Transcribed Image Text:uare, pomt 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? HEOREMS, 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 for practice. This section simply lays out a few basic principles and then those principles will be put to work in the rest of the course. The brief introduction provided in this section will not make you an instant expert at writing proofs, but it will equip you with the basic tools you need to get started. You should refer back to this section as necessary in the remainder of the course. Mathematical language An essential step on the way to the proof of a theorem is a careful statement of the theorem in clear, precise, and unambiguous language. To illustrate this point, consider the following proposition in incidence geometry. Proposition 2.5.1. Lines that are not parallel intersect in one point. Compare that statement with the following. Restatement. If l and m are two distinct lines that are not parallel, then there exists exactly one point P such that P lies on both l and m. Both are correct statements of the same theorem. But the second statement is much better, at least as far as we are concerned, because it clearly states where the proof should begin (with two lines l and m such that l + m and l f m) and where it should end (with a point P that lies on both l and m). This provides the framework within which we can
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