Question 1 At this point you might be asking yourself why it would be thought desirable to makemathematics so abstract and therefore to get into the kind of difficult issues that have beenraised here. That is one question we can answer. The answer is that abstraction is preciselywhat gives mathematics its power. By identifying certain key features in a given situation,listing exactly what it is about those features that is to be studied, and then studying themin an abstract setting detached from the original context, we are able to see that the samekinds of relationships hold in many apparently different contexts. We are able to studythe important relationships in the abstract without a lot of other irrelevant informationcluttering up the picture and obscuring the underlying structure. Once things have beenclarified in this way, the kind of logical reasoning that characterizes mathematics becomesan incredibly powerful and effective tool. The history of mathematics is full of examplesof surprising practical applications of mathematical ideas that were originally discoveredand developed by people who were completely unaware of the eventual applications.RCISES 2.41. It is said that Hilbert once illustrated his contention that the undefined terms in ageometry should not have any inherent meaning by claiming that it should be possibleto replace point by beer mug and line by table in the statements of the axioms. Considerthree friends sitting around one table. Each person has one beer mug. At the momentall the beer mugs are resting on the table. Suppose we interpret point to mean beermug, line to mean the table, and lie on to mean resting on. Is this a model for incidencegeometry? 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 axiomsdoes one-point geometry satisfy? Explain. Which parallel postulates does one-pointgeometry 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, interpretline to be a departmental committee, and interpret lie on to mean that the facultymember is a member of the specified committee.(a) Suppose the committee memberships are as follows: Alexander, Bailey, and Curtisare on the curriculum committee; Alexander and Dudley are on the personnelChapter 2Axiomatic Systems and Incidence Geometrycommittee; and Bailey and Curtis are on the social committee. Is this a model forIncidence Geometry? Explain.(b) Suppose the committee memberships are as follows: Alexander, Bailey and Curtisare on the curriculum committee; Alexander and Dudley are on the personnelcommittee; and Bailey and Dudley are on the social committee. Is this a model forincidence geometry? Explain.(c) Suppose the committee memberships are as follows: Alexander and Bailey are onthe curriculum committee, Alexander and Curtis are on the personnel committee,and Dudley and Curtis are on the social committee. Is this a model for incidencegeometry? Explain.4. A three-point geometry is an incidence geometry that satisfies the following additionalaxiom: 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. (Anaxiomatic 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 thesides of the square, and lie on to mean that the vertex is an endpoint of the side. Whichincidence axioms hold in this interpretation? Which parallel postulates hold in thisinterpretation?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 additionalaxiom: Every line has exactly three points lying on it. What is the minimum number ofpoints in such a geometry? Explain your reasoning.11. Find a finite model for Incidence Geometry in which there is one line that has exactlythree points lying on it and there are other lines that have exactly two points lying onthem.12. Find interpretations for the words point, line, and lie on that satisfy the followingconditions.(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.--:----- 1 .-d 2 Lald ..4 L .A --:----24

Given that we have to consider a beer mug as a point and table as a line. We know that we can get a…

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?