Use models to show that each of the following statements is independent of the axioms of...

Use models to show that each of the following statements is independent of the axioms of incidence geometry:

(a) Given any line, there are at least two distinct points that do not lie on it.

(b) Given any point, there are at least three distinct lines that contain it.

Solutions

Expert Solution

The axiom of incidence between points and lines are given below.

There are at least two distinct points .
There is one and only one line that contains two distinct points.
Every line contains at least two distinct points.

(a) Given any line, there are at least two distinct points that do not lie on it.

Consider the points A and B in the figure below. Both points are distinct points,but not on line. hence the given statement is independent of the axioms.

(b) Given any point, there are at least three distinct lines that contain it.

Consider the model given in figure below. Point A is a distinct point which contains three distinct lines.

Hence the statement is independent of axioms.

(c) Given any two distinct points, there is at least one line that does not contain either of them.

Consider the same figure of (a) as shown below. Here A and B are two distinct points. The line shown does not contains A and B. hence the statement is independent of axioms

Hope it helps.

Colby Messinger answered 1 year ago

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