Chin Ng

Math 300

Applications of Betweenness Axioms

Betweeness in an axiomatic system works as a valuable tool to indicate co-linearity and the distinction of points. Like other terms used in an axiomatic system, the idea of betweeness exists as an undefined term, but is a necessary concept used in logic to deduce ideas in interpretations and models. We’ll take an example of betweenness and our ability to deduce linearity and distinction of points with it as an illustration of a proof using the tool of betweenness.

Given that point B is between points A and C (written A*B*C) and A*C*D, we will prove that A, B, C and D are four distinct point and that they are all collinear. Pictorially, there are four points on a one-dimensional line where two of the points lie "between" the other two as shown here.

-------------A---------------B---------------C----------------D-----------

However, for a more accurate interpretation of what the above picture means, we must prove it through a base of facts (i.e. axioms). To do this, we must deny the conceptual possibilities of B=D and all four points (A, B, C, and D) failing to be collinear.

1) Since we know A*B*C and A*C*D, we start by noting that the set of points A, B and C are collinear and that the set of points A, C and D are collinear as well. This is stated in betweeness axiom 1: if A*B*C, then A, B and C are collinear.

2) Knowing that A, B and C are collinear, let’s say that they are incident with a line l. Having A and C lie on l means that D must lie on l as well since A, C and D are collinear. This illustration of collinearity informs us that A, B, C, and D all lie on line l; therefore, they are all collinear.

3) We now know that all four points are collinear; we also know that A¹ B, A¹ C, A¹ D, B¹ C and C¹ D through betweeness axiom 1 -- three points involved in betweenness are distinct from each other. In order to prove that all four points are distinct, we must also prove that B¹ D.

1. Consider a case where B=D. If B=D, then through substitution, we also know that A*D*C since we just replaced the B from A*B*C. This contradicts betweenness axiom 3, which states that given any three distinct points lying on the same line, then one and only one of them is between the other two. Having proven that A, D and C are three distinct, collinear points, we know that only one of those point can lie between the other two. Already having A*C*D, A*D*C cannot hold true. Thus, B¹ D and all four points (A, B, C and D) are distinct.

Through the illustration given above, we can deduce that, given A*B*C and A*C*D, the points A, B, C and D are distinct and collinear.

Bibliography

Greenburg, Marvin J. 1993. "Euclidean and Non-Euclidean Geometries," 3^rd edition, New York: W.H. Freeman and Company.