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19 Μαρ 2018 · A line is the set of points $(x,y)\in\mathbb{R}^2$ that satisfy the equation $$ax+by+c=0,$$ where at least one of $a$ or $b$ is non-zero. How to prove that given two distinct points $(p,q)$ and $(r,s)$, there is at most one line that contains both points?
- Number of points on a line in a finite projective plane
Every line contains at least 3 points. Small theorem: if b b...
- Number of points on a line in a finite projective plane
26 Ιαν 2016 · Every line contains at least 3 points. Small theorem: if b b and c c are distinct lines, there's a point that's on neither of them. Proof: The line b b intersects c c at some point Q Q by axiom B. Let B ≠ Q B ≠ Q be another point of b b (Axiom D), and C ≠ Q C ≠ Q be another point of c c. Consider the line d d containing B B and C C ...
18 Φεβ 2021 · Prove or disprove: \(2^n+1\) is prime for all nonnegative integer \(n\). Solution. Consider \(n=3;\) \(n\) is a nonnegative integer. \[2^n+1=2^3+1=9.\] \(9\) is not prime, since (\(3)(3)=9\), thus the statement: \(2^n+1\) is prime for all nonnegative integer \(n\) is false.
First, a line contains infinitely many points. The idea here is that if you have two distinct lines which intersect, there is only one (unique) plane that contains both lines and all of their points. Try visualizing a plane that contains two intersecting lines:
It is easy to find values for x and z satisfying the first, such as x = 1, z = 0 and x = 2, z = 1. Then we can find corresponding values for y using the second equation, namely y = 3 and y = 1, so (1, 3, 0) and (2, 1, 1) are both on the line of intersection because both are on both planes.
It is not hard to prove that a FPP contains points; every line has n + 1 points, every point is on n + 1 lines, and n is at least 2. (See problems 7 – 11). n2 +n +1 This number n is called the order of the FPP. 5. Finite Affine Planes. Axioms: (F)AP-1. Every two distinct points determine a unique line. (F)AP-2.
13 Ιουλ 2021 · In a finite affine plane, if one line contains exactly \(n\) points, then every line contains exactly \(n\) points. Proof. Again, we begin with a diagram that may be a helpful visual aid to understanding this proof. Let \(L\) be a line that contains exactly \(n\) points, and let \(L'\) be any other line of the finite affine plane.
