Question

Let n be an integer with n≥2. A lineal geometry on n points is an axiomatic system whose undefined terms are point, line, and lies on (and similar terms such as contains) and whose axioms are: Lineal Axiom 1 Every line contains exactly two points, and no point lies on more than two lines: Lineal Axiom 2 There are exactly n points. Moreover, there are exactly n−2 points which lie on two distinct lines, and there are exactly 2 points each of which is contained in exactly one line. Lineal Axiom 3 If all n points are partitioned into two sets of points A and B such that A∩B=∅ and neither A nor B is empty, then there is at least one point X in A and one point Y in B such that X and Y lie on the same line. (a) For each of n=2,n=3,n=4,n=5, and n=6, depict a lineal geometry on n points using dots and arcs.

Answer 1

A lineal geometry on n points is an axiomatic system that follows certain rules regarding the relationship between points and lines. You can represent lineal geometries for n=2, n=3, n=4, n=5, and n=6 using dots and lines.

Step-by-step explanation:

A lineal geometry on n points is an axiomatic system with 2 undefined terms, point and line, and 3 axioms. The axioms state that every line contains exactly two points, and no point lies on more than two lines; there are exactly n points, with n-2 points lying on two distinct lines, and 2 points each contained in exactly one line; and if all n points are partitioned into two sets, then there is at least one point in each set that lies on the same line.

For n=2, you can represent a lineal geometry with 2 points as two dots connected by a line. For n=3, you can have 3 points with two dots connected by a line and one dot that is not connected to any other. For n=4, you can have 4 points with two separate lines, each containing two dots. For n=5, you can have 5 points with three lines, one with three dots and two with one dot each. For n=6, you can have 6 points with three lines, each containing two dots.