Question 8, i've attached examples 3 and 4

sting 8. Show that the interpretations in Examples 3 and 4 of this chapter 12. (a) Show are models of incidence geometry and that the Euclidean and hy- 1# 1 perbolic parallel properties, respectively, hold for them. (b) Show 9. In each of the following interpretations of the undefined terms. affin which of the axioms of incidence near- every EXAMPLE 3. Let the "points" be the four letters A, B, C, and D. Let the "lines" be all six sets containing exactly two of these letters; {A, B}, {A,C}, {A, D}, {B, C), {B, D}, and {C, D}. Let "incidence" be set membership, as in Example 1. As an exercise, you can verify that this is a model for incidence geometry and that in this model the Euclidean parallel postulate does hold (see Figure 2.5). By Examples 1 and 3, the Euclidean parallel postulate is independent of the axioms of incidence geometry. EXAMPLE 4. Let the "points" be the five letters A, B, C, D, and E. Let the "lines" be all 10 sets containing exactly two of these letters.MODELS 75 Figure 2.5 Euclidean parallel property. A four-point incidence geometry. Let "incidence" be set membership, as in Examples 1 and 3. You can verify that in this model the following statement about parallel lines, called the hyperbolic parallel property, holds: "For every line I and every point P not on l, there exist at least two lines through P parallel to !" (see Figure 2.6). (The figures illustrating Examples 1, 3, and 4 are only meant to be suggestive. They have features not included in the definition of those models in terms of letters. For example, in Figure 2.5, the dash illus- trating the "line" {A, D} appears to intersect the dash illustrating the "line" {B, C} when those "lines" are actually parallel, so it's better to view Figures 2.5 and 2.6 as three-dimensional drawings.) Let us summarize the significance of models. Models can be used to demonstrate the impossibility of proving or disproving a statement from the axioms. We just showed the undecidability of the Euclidean, elliptic, and hyperbolic statements about parallel lines in incidence geometry. Moreover, if an axiom system has many models that are es- sentially different from one another, as are the models in Examples 1, 3, and 4, then that system has a wide range of applicability. Proposi- tions proved from the axioms of such a system are automatically