Primitive Notions and Propositions

Author:: Jorge Cássio

Topic:: Straight Lines, Planes

What is an axiom?

Axioms (or postulates) are unquestionably universally valid truths, often used as principles in the construction of a theory or as the basis for an argument. That is, an axiom is a proposion which is so clear, that is assumed as true without a demonstration or proof. An axiomatic system is the set of axioms that define a given theory and that constitute the simplest truths from which the new results of that theory are demonstrated. What this means is that for every theorem in math, there exists an axiomatic system that contains all the axioms needed to prove that theorem. The axiom contains evidence in itself and therefore does not need not be proved.

Incidence axioms

Axiom 1- Whatever the line, there are points that belong to this line, and points not belonging to it.

Axiom 2- Given two distinct points, there is a single line that contains them. Note: when two lines have a point in common, they are said to intersect or cut at that point.

Activity 1

In the previous Geogebra applet, select the Line option (window 3) and draw a line that passes through points B and D. Draw another line that passes through D and C. Is this last line in the same plane of the other two lines?

Proposition 1

Two distinct lines either do not intersect or intersect at a single point.

Activity 2

Which pairs of lines intersect? Do lines g and h intersect?

Proof of Proposition 1

Hypothesis: Consider m and n as two distinct lines. Thesis: m and n do not intersect or intersect at a single point. The intersection of these two lines cannot contain two or more points, otherwise, when looking at the truth provided by axiom 2 (Given two distinct points there is a single line that contains them), they would coincide. Therefore the intersection of m and n either happens at only one point or it doesn't happen.

Activity 3

Did you understand the proof? Can you explain it in another way? If you didn't understand, write what you didn't understand.

Axiom 3

Given three distinct points on a line, one and only one of them is located between the other two.

Definition 1: Segment

The set consisting of two points A and B and all points between A and B is called line segment AB. Points A and B are called line segment endpoints.

Activity 4

In the previous construction, select the Segment tool (window 3) and create the line segments BC, CE and AC. If you didn't understand, write what you didn't understand.

Definition 2: Ray

Consider A and B as two distinct points. The set consisting of points of segment AB and of all points C so that B is between A and C, is called a ray starting at A passing through B.

Note 1

In reference books, in general, the ray is represented only with an "Vector".

Note 2

Note that two points A and B determine two rays: S_ABand S_BA.

Axiom 4

Consider two points A and B of a line. There is always a point C between A and B , and a point D so that B is between A and D.

Illustration

Note 3

The result in this axiom is that, between any two points on a line, there is an infinity of points.

Definition 3: Half-plane

Consider m as a line and A a point that does not belong to line m. The set consisting of m and all points B, such that A and B are on the same side of line m, is called a half-plane determined by m containing A.