Axiomatic System Undefined Terms Technical Terms : Definitions defined within these Undefined Terms Set of statements(Axioms) dealing with Undefined Terms and definitions All other Statements of the system- Theorems Logical consequences of these axioms.
Independent An axiom is said to be independent If it can not be logically deduced from the other axioms in the system.
Completeness A set of axioms is said to be complete If it is not possible to add any independent axiom on the system.
Consistency A set of axioms is said to be consistent If it is impossible to deduce a theorem (from axioms) that contradict any axiom or previously proved theorem.
Euclidean Geometry The Euclidean Geometry is an axiomatic system along with five axioms . The five axioms are: 1. To draw a straight line from any point to any point. 2. To produce [extend] a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance [radius]. 4. That all right angles are equal to one another. 5. The parallel postulate: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. It is possible to reject any or all of these and still have valid geometries with different principles. For example, if we simply toss out #5, we are no longer dealing with Euclidean Geometry, we are dealing with Hyperbolic geometry (in which there are infinite distinct lines parallel to another line through a given point)…. And if …
GeometryEuclidean Geometry Neutral Geometry Non-Euclidean Geometry there are infinite distinct lines parallel Axiom:-5 through a given point there is unique line Common parallel through a given point Axioms …… Axiom:-5 1,2,3,4 there are no lines parallel through a given point
Four Point GeometryUndefined Terms: Point, Line, On. Axiom 1: There exist exactly four distinct points. Axiom 2: Any two distinct points have exactly one line on both. Axiom 3: Each line is on exactly two points.Def 1: Two lines on the same point are said to intersectDef 2: Two lines that do not intersect are called parallel.Theorem 1: Each point of the four-point geometry has exactly three lines on it…
Are blue lines parallel?Complete four point Yes, see definition 2