MAT 360 Lecture 5

Hilbert’s axioms - Betweenness

2

EXERCISE:

Can you deduce from the Incidence Axioms that there exist one point and one line?

Can you deduce from the Euclid’s I to V Axioms that there exist one point and one line? 2

Incidence Axioms1. For each point P and for each point Q not

equal to P there exists a unique line incident with P and Q.

2. For every line T there exist at least two distinct points incident with T.

3. There exist three distinct points with the property that no line is incident with all the three of them.

Euclid’s postulatesI. For every point P and every point Q not

equal to P there exists a unique line l that passes for P and Q.

II. For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and the segment CD is congruent to the segment BE.

III. For every point O and every point A not equal to O there exists a circle with center O and radius OA

IV. All right angles are congruent to each other

V. For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.

Hilbert’s Axioms

IncidenceBetweennessCongruenceContinuityParallelism

Note: you need to read all Chapter 3 while we work on it. Every statement previously proved in the text can be used

Notation

By

A*B*C we will mean “the point B is between the point

A and the point C.”

AXIOMS OF BETWEENNESS (first part) B1: If A*B*C then A, B and C are three

distinct points lying on the same line and C*B*A.

B2: Given two distinct points B and D, there exist points A, C and E lying on BD such that A*B*D, B*C*D and B*D*E.

B3: If A, B and C are distinct points lying on the same line, then one and only one of the points is between the other two.

EXERCISES

Write the axiom B3 using the notation * we’ve just introduced.

Can you find a model for the Betweeness Axioms?

Consider a sphere S in Euclidean three-space and the following interpretation: A point is a point on S, a line is a great circle on S and incidence is set membership. Is this intrepretation a model of Betweeness Axioms? (What about Incidence Axioms?)

Old definitions revisted

The segment AB is the set of all points C such that A*C*B together with the points A and B.

The ray AB is the set of points on the segment AB together with all the points C such that A*B*C.

EXERCISE

Let A and B denote two points. Prove that AB ∩ BA = AB AB U BA = AB

Definition

Let l be a line. Let A and B be points not lying on l. We say that A and B are on the same side of l if

A=B or the segment AB does not intersect l. We say that A and B are on opposite sides of l

if A ≠ B and the segment AB does intersect l.

Questions

Suppose you have two points A and B lying on a line l. Are A and B on the same side of l or on opposite

sides of l? Suppose you have two points A lying on a

line l and B not lying on l. Are A and B on the same side of l or on opposite

sides of l?

AXIOMS OF BETWEENNESS (second part) B4: For every line l and for every three points

A, B and C not lying on l, If A and B are on the same side of l and B and C

are on the same side of l, then A and C are on the same side of l.

If A and B are on opposite sides of l and B and C are on opposite sides of l then A and C are on the same side of l.

Proposition

If A and B are on opposite sides of l and B and C are on same side of l then A and C are on opposite sides of l.

Definition:

A set of points S is a half plane bounded by a line l if there exists a point A such that S consists in all the points B for which A and B are on the same side of l.

Propositions

Every line bounds exactly two half planes and these two have planes have no point in common.

If A*B*C and A*C*D then B*C*D and A*B*D. If A*B*C and B*C*D then A*B*C and A*C*D (line separation property) If C*A*B and l is

the line through A, B and C then for every point P lying on l, P lies either on the ray AB or on the ray AC

Pasch Theorem

If A, B and C are distinct noncollinear points and l is any line intersecting the line AB in a point between A and B, then l intersects either AC or BC. If C does not lie on l then l does not intersect both AC and BC.

Proposition

If A*B*C then B is the only point lying on the rays BA and BC and AB=AC.

Definition

A point D is in the interior of an angle <CAB if D is on the same side of the line AC as B and D is on the same side of the line AB as C.

Definition

The interior of a triangle is the intersection of the interior of its three angles.

A point P is exterior to a triangle if it is not an interior point of a triangle and does not lie in any side of the triangle.

Proposition

If D is in the interior of <CAB then Every point in the ray AD except A is in the interior

of <CAB None of the points in the ray opposite to the ray

AD are in the interior of <CAB If C*A*E then B is in the interior of <DAE

Definition

Ray AD is between rays AC and AB if AB and AC are not opposite rays and D is interior to <CAB.

Crossbar theorem

If the ray AD is between rays AC and AB then AD intersects segment BC

EXERCISE (18, Chapter 3) Consider the following interpretation. Points: points (x,y) in

the Euclidean plane such that both coordinates, x and y, have the form a/2n

Lines: Lines passing through several of those points.

Show that The incidence axioms

hold The first three

betweenness axioms hold.

Line separation property holds.

Pasch theorem fail What about Crossbar

theorem?