Index
Let's continue with our discussion of axioms systems. An axiom system is considered dependant if one or more of the axioms follows immediately (or can be proven) from the other axioms. It is an independant axiom system otherwise. To show that an axiom system is dependant, it is only necessary to show that one axiom follows from the others. However to show that an axiom system is independant, one must show that no one axiom follows from the others. One can do this by showing that, for any given axiom in the axiom system, there is a geometry that satisfies any subset of the remaining axioms along with the negation of the chosen axiom. For example, if the axiom system as 5 axioms:
- Axiom 1
- Axiom 2
- Axiom 3
Then one would have to find a geometry for each of the following axioms systems (the symbol ~ refers to the operation of negation):
- Axiom 1, ~Axiom 3
(this shows that Axiom 3 does not follow from Axiom 1)
- Axiom 2, ~Axiom 3
(this shows that Axiom 3 does not follow from Axiom 2)
- Axiom 1, Axiom 2, ~Axiom 3
(this shows that Axiom 3 does not follow from Axioms 1 and 2)
- Axiom 1, ~Axiom 2
(this shows that Axiom 2 does not follow from Axiom 1)
- Axiom 3, ~Axiom 2
(this shows that Axiom 2 does not follow from Axiom 3)
- Axiom1, Axiom 3, ~Axiom 2
(this shows that Axiom 2 does not follow from Axioms 1 and 3)
- Axiom 2, ~Axiom 1
(this shows that Axiom 1 does not follow from Axiom 2)
- Axiom 3, ~Axiom 1
(this shows that Axiom 1 does not follow from Axiom 3)
- Axiom 2, Axiom 3, ~Axiom 1
(this shows that Axiom 1 does not follow from Axioms 2 and 3)
As you can see, it can be difficult to show that a given axiom system is independant. Do you think that you can have an axiom system that is both independant and inconsistent?
Let's now look at some more well known geometries. One of particular interest is the axiom system that defines Projective Planes.
PROJECTIVE PLANES
1. Two distinct points are contained in a unique line.
2. Two distinct lines intersect at a unique point.
3. There exist four points of which no three are incident with the same line.
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PROBLEM 2:
(a) Construct an example that satisfies the three axioms of a projective plane. (HINT: Think about how many points your example should contain to satisfy these axioms. You will need to use a circle as one of your lines!)
Here are some more geometries:
AFFINE PLANES
1. Two distinct points are contained in a unique line.
2. Given a line and a point, there is a unique line through the point that is non incident with, or "parallel" to, the given line.
3. There exist three points that are not all contained in a line.
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LINEAR SPACES
1. Any line has at least two points.
2. Two points are on at most one line.
3. Any two points are on a line.
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DESIGNS
1. Each line contains at least two points.
2. Two points are on at least one line.
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ABSTRACT PLANE CONFIGURATIONS
1. There are p points and l lines with n lines through every point and m points on every line.
2. Two distinct points are contained in at most one line.
3. Two distinct lines interstect in at most one point.
4. The geometry is connected.
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GENERALIZED QUADRANGLE
1. Two distinct points are contained in at most one line.
2. Given a line l and a point p not on l, there is exactly one line k through p that intersects l (in some point q ).
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BIPLANES
1. Two distinct points are contained in two distinct lines.
2. Two distinct lines intersect in two distinct points.
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(b) Construct an example for each of these geometries.
Solutions
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Notice how each of the geometries introduced are defined by certain incidences. Nearly all of these can be rewritten using the incidence terminology referred to in the introduction. The incidence of lines and points is very important in defining geometries.
References: Batten, Polster.
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