Definition. An incidence structure consists of a set of objects P called \"point

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Definition. An incidence structure consists of a set of objects P called "points", and a second set of objects L called "lines". There is a single relation called "lies on" which associates to each point a subset of lines that it lies on. When this is the case, where PEP, le L and P lies on l, as you would expect we write Pel. For example, the usual points and lines in R™ from an incidence structure. But there are many others. For example the set of cookies in my kitchen could be taken as points and the set of cookie containers could be taken as lines. The lies on relation could be chosen to be true if a cookie lies inside a container. This incidence structure may not be very interesting from the point of view of Geometry. but this example illustrates the freedom we have in specifying incidence structures. The reason for studying incidence structures is that many mathematical problems can be solved by using them, particularly geometric questions. Two other incidence structures will be important to us: the points and lines in the hyperbolic plane H2 (we will use the Poincare disc model), and the points in lines in the Single Point Elliptic Geometry S2 (as in the disc model), where for now we will ignore distance. In order to study incidence structures we have to be careful about how we use language. We will allow the logical connectives common in mathematics (and, or, if-then, not) as well as the quantifiers (for all, there exists). As we have in this course, we will use upper case letters for points, and lower case letters for lines. We also have the "lies on" relation for which we use the symbol e. We build sentences of incidence geometry from these language elements. For example, we can say that two lines li and ly are parallel lines in this language as follows: Both li EL and l2 L, and there does not exits PEP with e l and Pel2. This may read somewhat clumsy and we won't always write out every sentence this formally, but we need to know that when we are discussing incidence geometry the notions we are discussing can be formulated this way. A sentence of incidence geometry can be true in some structures and false in others. For example the parallel postulate is a sentence in incidence geometry: it is true in R2 but false in H2 and S2. It is the existence of these structures that enables us to know that the parallel postulate can never be proved using Euclid's first four axioms, since they are true in the later two structures. The number of incidence geometries is endless. So we restrict our attention to two types. Each is specified by its own set of Axioms. Axioms of Affine Geometry: A1. There is a unique line containing any two distinct points. A2. If P is a point not on a line l, there there is a unique line containing P parallel to l (meaning this line and l have no points in common.) A3. There exist three non collinear points. Axioms of Projective Geometry: P1. There is a unique line containing any two distinct points. P2. Any two distinct lines intersect in a unique point.

Explanation / Answer

Let us see what is duality.

In maths, duality means obtaining a new expression from all ready existing meaningfull expression by simply interchanging the words.

Euclid's first four axioms are

1 Things which are equal to the same thing are also equal to one another.

2 If equals are added to equals, the whole are equal.

3 If equals be subtracted from equals, the remainders are equal.

4. Things which coincide with one another are equal to one another.

Hence proved

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