DIRECTIONS: For each of the following questions, you may only site propositions 6.1, 7.1, 7.2, and 7.3; the incidence axioms, the betweenness axioms, and any definition from sections 6 & 7 of Hartshorne's textbook. You may also use exercises from the textbook, but only if you (or we as a class) have proved them and their number comes before whatever exercise you are trying to use it to prove. 1. (Exercise 7.5 - adjusted) Consider the model: the ring where points are {0,1,2,3,4}. We define points A, B, C to be between in the sense that A*B*C if we have B = {(a + c)(mod5). (Hint: we talked about this model in class. Additional information about mods is on the course's canvas page.] (a) Show that the axioms (B1), (B2) and (B3) hold in this model. (b) Prove that Line Separation (prop 7.2) does not hold in this model by giving an example where line separation fails. (c) Based on the previous parts, explain why Line Separation can not be a consequence of axioms (B1), (B2), & (B3). DIRECTIONS: For each of the following questions, you may only site propositions 6.1, 7.1, 7.2, and 7.3; the incidence axioms, the betweenness axioms, and any definition from sections 6 & 7 of Hartshorne's textbook. You may also use exercises from the textbook, but only if you (or we as a class) have proved them and their number comes before whatever exercise you are trying to use it to prove. 1. (Exercise 7.5 - adjusted) Consider the model: the ring where points are {0,1,2,3,4}. We define points A, B, C to be between in the sense that A*B*C if we have B = {(a + c)(mod5). (Hint: we talked about this model in class. Additional information about mods is on the course's canvas page.] (a) Show that the axioms (B1), (B2) and (B3) hold in this model. (b) Prove that Line Separation (prop 7.2) does not hold in this model by giving an example where line separation fails. (c) Based on the previous parts, explain why Line Separation can not be a consequence of axioms (B1), (B2), & (B3)