using these axioms ( finite affine plane ) : AA1 : there exists at least 4...

using these axioms ( finite affine plane ) :

AA1 : there exists at least 4 distinct points , no there of which are collinear .

AA2 : there exists at least one line with n (n>1) points on it .

AA3 : Given two distinct points , there is exactly one line incident with both of them .

AA4 : Given a line l and a point p not on l , there is exactly one line through p that does not intersect l .

prove : in an affine plane of order n , each line contains exactly n points .

Expert Solution

milcah answered 2 years ago

Given the following Axioms of Fano's geometry: 1. There exists at least one line 2. Each...

Given the following Axioms of Fano's geometry: 1. There exists at least one line 2. Each line is on exactly 3 points 3. Not all points are on the same line 4. Each pair of points are on exactly one line 5. Each pair of lines are on at least one point a) Prove every point is on exactly three lines b) What geometries are possible if you eliminate Axiom 5?

Assign the coordinates to each point on an Affine plane of order 3.

A projective plane is a plane (S,L ) satisfying the following four axioms. P1. For any...

A projective plane is a plane (S,L ) satisfying the following four axioms. P1. For any two distinct points P and Q there is one and only one line containing P and Q. P2. For any two distinct lines l and m there exists one and only one point P belonging to l∩m. P3. There exist three noncollinear points. P4. Every line contains at least three points. Let π be a projective plane. Using P1 − P4, show that π...

Please justify and prove each statement (Use explicitly the four axioms) a) Prove that a finite...

Please justify and prove each statement (Use explicitly the four axioms) a) Prove that a finite positive linear combination of metrics is a metric (Use explicitly the four axioms). If it is infinite, will it be metric? b) Is the difference between two metrics a metric? (d1 - d2)

Q3 (a) Briefly describe the extended Euclidean plane, and explain how it satisfies the axioms of...

Q3 (a) Briefly describe the extended Euclidean plane, and explain how it satisfies the axioms of projective geometry. (b) Describe the Real Projective Plane RP2, explaining what points and lines in RP2 correspond to in R 3 with respect to an ‘all-seeing eye’ at the origin. (c) Prove that the Cross-Ratio is preserved by every projective transformation

8. Definition: A set A is finite if there exists a non-negative integer c such that...

8. Definition: A set A is finite if there exists a non-negative integer c such that there exists a bijection from A to {n ∈ N : n ≤ c}. (The integer c is called the cardinality of A.) (a) Let A be a finite set, and let B be a subset of A. Prove that B is finite. (Hint: induction on |A|. Note that our proof can’t use induction on |B|, or indeed refer to “the number of elements...

Using field axioms and order axioms prove the following theorems (explain every step by referencing basic...

Using field axioms and order axioms prove the following theorems (explain every step by referencing basic axioms) (i) The sets R (real numbers), P (positive numbers) and [1, infinity) are all inductive (ii) N (set of natural numbers) is inductive. In particular, 1 is a natural number (iii) If n is a natural number, then n >= 1 (iv) (The induction principle). If M is a subset of N (set of natural numbers) then M = N The following definitions...

Consider n-dimensional finite affine space: F(n,p) over the field with prime p-elements. a. Show tha if...

Consider n-dimensional finite affine space: F(n,p) over the field with prime p-elements. a. Show tha if l and l' are two lines in F(n,p) containing the origin 0, then either l intersection l' ={0} or l=l' b. how many points lie on each line through the origin in F(n,p)? c. derive a formula for L(n,p), the number of lines through the origin in F(n,p)

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an...

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an operator T on V such that the null space of T is U.

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F...

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F = 0_F . Show further that the smallest positive integer with this property is a prime number.

Question