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Chapter 2: Problem 1

Using the axioms of incidence and betweenness and the line separation property, show that sets of four points \(A, B, C, D\) on a line behave as we expect them to with respect to betweenness. Namely, show that (a)\(A * B * C\) and \(B * C * D\) imply \(A * B * D\) and \(A * C * D\) (b)\(A+B * D\) and \(B * C * D\) imply \(A * B * C\) and \(A * C * D\)

Short Answer

Expert verified
For part (a), both \(A * B * D\) and \(A * C * D\) result from the given conditions. For part (b), given the conditions, \(A * B * C\) and \(A * C * D\) are the derived results.

Step by step solution

01

Explanation of Symbols

Here the symbols have the following meanings: \(*\) symbol is used to denote the betweenness relation (e.g., \(A*B*C\) meaning \(B\) is between \(A\) and \(C\)), and \(+\) symbol denotes existence of a connection(in this context), e.g., \(A+B\) meaning there exists a connection between \(A\) and \(B\).
02

Solving for part (a)

Given \(A * B * C\) and \(B * C * D\). From the first condition, \(B\) is between \(A\) and \(C\), and from the second, \(C\) is between \(B\) and \(D\). From these statements it can be deduced that \(A * B * D\) since \(B\) is between \(A\) and \(D\) (i.e. \(B\) lies between \(A\) and \(C\) and \(C\) lies between \(B\) and \(D\)) and \(A*C*D\) since \(C\) is between \(A\) and \(D\)(i.e. \(C\) lies between \(B\) and \(D\) and \(B\) lies between \(A\) and \(C\)).
03

Solving for part (b)

Given \(A + B * D\) and \(B * C * D\). This means that \(A\) and \(B\) are connected in some form, and \(C\) is between \(B\) and \(D\). Based on these conditions it can be concluded that \(A * B * C\) and \(A * C * D\), because there is a connection between \(A\) and \(B\) and since \(C\) is between \(B\) and \(D\), \(B\) should be between \(A\) and \(C\) and \(C\) should be between \(A\) and \(D\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Axioms of Incidence
The axioms of incidence are fundamental rules in geometry that describe how points, lines, and planes relate to each other. These axioms establish that points lie on lines or planes, and lines lie on planes. Here we look at a basic incidence axiom that says any two points, like points A and B, are connected by exactly one line. This axiom is crucial for defining the concept of betweenness because it allows us to consider the idea of one point being between two others on a line.

Understanding this principle is important when dealing with sets of four points like in our exercise. It ensures that when we talk about points A, B, C, and D lying on the same line, we're working within a framework where the existence of such a line isn't just assumed, but guaranteed by the rules of geometry.
Line Separation Property
The line separation property in geometry gives us a way to talk about the arrangement of points on a line. It states that if a point B lies between points A and C on a line, that line is divided or 'separated' into two distinct parts. Therefore, point B is not only painted in between A and C, but it also separates them. This concept is fundamental when discussing betweenness as it helps us understand how points divide lines into segments.

In our exercise, when we know that point B lies between A and C (A * B * C), and point C lies between B and D (B * C * D), we can deduce how points A, B, C, and D are separated along the line. The line separation property gives a precise meaning to 'betweenness' and is key to solving the problem at hand.
Geometric Betweenness
Geometric betweenness is a way of describing the relative positions of points on a line. When we say B is between A and C (written as A * B * C), we mean that B lies on the line segment that connects A and C, and traveling from A to C along the line, one would encounter B before reaching C. The concept of betweenness helps us understand the notion of order for points on a line.

In relation to the exercise, when certain points are known to be between others, it allows us to infer additional relationships among points on the line. For example, knowing that A * B * C indicates that B is closer to C than A is to C. It's this principle that facilitates the solution to parts (a) and (b) of the problem, where understanding the order of points and the relationship of betweenness is crucial to determining the placement of points on the line.

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Most popular questions from this chapter

A subset \(W\) of the plane is segment-connected if given any two points \(A, B \in W\), there is a finite sequence of points \(A=A_{1}, A_{2}, \ldots, A_{n}=B\) such that for each \(i=1,2, \ldots\) \(n-1,\) the segment \(A_{i} A_{i+1}\) is entirely contained within \(W\) If \(A B C\) is a triangle, show that the exterior of the triangle, that is, the set of all points of the plane lying neither on the triangle nor in its interior, is a segment-connected set.

The following exercises (unless otherwise specified) take place in a geometry with axioms ( 11 ) - ( 13 ), ( B1 ) - (B4), (C1)-(C3). Consider the real Cartesian plane \(\mathbb{R}^{2}\), with lines and betweenness as before (Example 7.3 .1 ), but define a different notion of congruence of line segments using the distance function given by the sum of the absolute values: $$d(A, B)=\left|a_{1}-b_{1}\right|+\left|a_{2}-b_{2}\right|$$ where \(A=\left(a_{1}, a_{2}\right)\) and \(B=\left(b_{1}, b_{2}\right) .\) Some people call this "taxicab geometry" because it is similar to the distance by taxi from one point to anther in a city where all streets run east-west or north-south. Show that the axioms (C1), (C2), (C3) hold, so that this is another model of the axioms introduced so far. What does the circle with center ( \(0,0)\) and radius 1 look like in this model?

The following exercises (unless otherwise specified) take place in a geometry with axioms ( 11 ) - ( 13 ), ( B1 ) - (B4), (C1)-(C3). Let \(r\) be a ray originating at a point \(A\) and let \(s\) be a ray originating at a point B. Show that there is a 1 -to-1 mapping \(\varphi: r \rightarrow s\) of the set \(r\) onto the set \(s\) that preserves congruence and betweenness. In other words, if for any \(X \in r\) we let \(X^{\prime}=\varphi(X) \in s,\) then for any \(X, Y\) \(Z \in r, \quad X Y \cong X^{\prime} Y^{\prime}, \quad\) and \(\quad X * Y * Z \Leftrightarrow\) \(X^{\prime} * Y^{\prime} * Z^{\prime}\)

Construct with Hilbert's tools a line perpendicular to a given line \(l\) from a point \(A\) not on \(l(\text { par }=4)\).

A set \(U\) of points in the plane is a convex set if whenever \(A, B\) are distinct points in \(U\) then the segment \(\overline{A B}\) is entirely contained in \(U\). Show that the inside of a triangle is a convex set.
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