Modern Geometry: Test 1 Solutions for Affine and Projective Planes - Prof. Ma Filaseta, Exams of Mathematics

Download Modern Geometry: Test 1 Solutions for Affine and Projective Planes - Prof. Ma Filaseta and more Exams Mathematics in PDF only on Docsity! MATH 532, 736I: MODERN GEOMETRY Test 1, Spring 2011 Name Show All Work Instructions: Put your name at the top of this page and at the top of the first page of the packet of blank paper given to you. Work each problem on the paper provided, using a separate page for each problem. Show ALL of your work. Put your answers in the boxes below where appropriate. Do NOT use a calculator. Points: Part I (52 pts), Part II (48 pts) Part I. The point value for each problem appears to the left of each problem. In Problem 5, I will assume you are using the axioms as you state them in your answer to Problem 1 below. (1) In the packet of white paper provided to you, state the axioms for a finite AFFINE plane of order n. (Number or name the axioms so you can refer to them in Problem 5.) 12 pts (2) In the packet of white paper provided to you, give a model for a finite PROJECTIVE plane of order 3. Be sure to clearly mark every point and clearly draw every line in your model. 8 pts (3) Two points have been circled in the 7× 7 array of points to the right. Using the model for a finite affine plane of order 7 discussed in class, finish circling the points that belong to the same line as the given circled points. 8 pts (4) Consider the points (4, 2) and (10, 9) in an 11 × 11 array of points for our model of a finite affine plane of order 11. Find the equation of the line passing through these two points. Put your answer below in the form y ≡ mx+k (mod 11) where m and k are among the numbers 0, 1, 2, . . . , 10. Be sure to show your work in the packet provided with this test. Answer: 8 pts (5) Using only the theorem below and the axioms you stated in Problem 1, fill in the boxes below to complete a proof that in an affine plane of order n, for each line `, there are at least n − 1 lines parallel to `. This is part of a proof you were to have memorized for class. (More precisely, the problem you were to have memorized for class involved showing that there are “exactly” n−1 such lines. I have not used the word “exactly” in my statement of the problem above.) Theorem. In an affine plane of order n, each line contains exactly n points. Note: The theorem is to be used in the proof below. The proof is establishing that there are at least n− 1 lines parallel to ` as stated above. Proof. Let ` be an arbitrary line. By , there is a point P1 not on `. By , line ` has at least one point, say P2, on it. From , there is a line `′ passing through P1 and P2. Since , we have that `′ 6= `. implies that P2 is the only point on both `′ and `. By , `′ has exactly points on it, two of which are P1 and P2. Let P3, . . . , Pn denote the remaining points on `′. For each j 6= 2, Pj is not on ` so that implies that there is a line `j passing through Pj and parallel to `. Each such `j is different from `′ since . It follows that the lines `j (with j 6= 2) are distinct by (since `′ is the unique line passing through any two of the Pj’s). Thus, there are at least n−1 distinct lines parallel to ` (namely, the lines `j with j 6= 2).  16 pts (3) Two points have been circled in the 7x7 array of points to the right. Using the model for a finite affine plane of order 7 discussed in class, finish circling the points that belong to the same line as the given circled points. (4) Consider the points (4, 2) and (10, 9) in an 11x11 array of points for our model of a finite affine plane of order 11. Find the equation of the line passing through these two points. Put your answer in the form a. Find the slope of the line by using the slope formula: You only need to use one point to find the k. However, for information sake, both are shown. By the slope formula given above, we find m= 7/6. Note that 7 = 18 (mod 11) so we can use m = 18/6 = 3 for the slope. 𝑦 3𝑥 𝑘 2 3 4 𝑘 2 2 𝑘 2 − 2 𝑘 − 0 𝑘 𝑘 𝑦 3𝑥 a) Using the point (4, 2) 𝑦 3𝑥 𝑘 9 3 0 𝑘 9 30 𝑘 9 − 30 𝑘 −2 𝑘 𝑘 𝑦 3𝑥 b) Using the point (10, 9) Definition: if 𝑎 𝑏 𝑚 , the m divides (a-b). So we know that, −2 𝑘 ⟹ 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 2 𝑘 ⟹ 𝑘 , since k is in {0,1,...,10} (5) Using only the theorem below and the axioms you stated in Problem 1, fill in the boxes below to complete a proof that in an affine plane of order n, for each line , there are at least n-1 lines parallel to . This is part of a proof you were to have memorized for class. (More precisely, the problem you were to have memorized for class involved showing that there are exactly n-1 such lines. I have not used the word “exactly” in my statement of the problem above.) Theorem: In an affine plane of order n, each line contains exactly n point. Note: The theorem is to be used in the proof below. The proof is establishing that there are at least n-1 lines parallel to as stated above. Proof: Let be an arbitrary line. By Axiom A₁, there is a point P₁ not on . By the given Theorem, line has at least one point, say P₂, on it. From Axiom A₃, there is a line passing through P₁ and P₂. Since P₁ is on and not on , we have that . Axiom A₃ implies that P₂ is the only point on both . By the given Theorem, has exactly n points on it, two of which are P₁ and P₂. Let P₃, …, denote the remaining points on. For each 2, is not on so that Axiom A₄ implies that there is a line passing through and parallel to . Each such is different from since is parallel to and is not. It follows that the lines (with 2) are distinct by Axiom A₃ (since is the unique line passing through any two of the ). Thus there are at least n-1 distinct line parallel to (namely, the lines with 2). Part II: The problems in this section all deal with an axiomatic system consisting of the following axioms. Axiom 1. There exist 3 collinear points (that is, 3 points and a line with the 3point on the line). Axiom 2. There exist exactly 3 distinct lines. Axiom 3. Given two distinct lines, there is at least one point on both lines. Axiom 4. Given two distinct points, there is at most one line passing through them. Note: Axiom 1 is saying that there exist 3 collinear points. This does NOT mean “exactly”. There may be more points in the axiomatic system, and there may even be more point on the same line as these 3 points. 1) Justify that the axiomatic system is consistent. 2) Justify that the axiomatic system in not compete. Include some brief explanation for your answer. The model satisfies all the axioms in the system, therefore the axiomatic system is consistent NOTE: this is just one model that satisfies the axioms. The axiomatic system is not complete because there exist more than 1 model for the system, and it is possible for a theorem about the system to hold for one model but not for another. Ie: in the model on the left there exist 7 points which is not true for the model on the right. NOTE: Because we can add at least one point to at least one line when constructing our model , implies that we can have as many points as desired. Although we need only show two, there are infinitely many models.