Download - MAT 360 Lecture 5
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MAT 360 Lecture 5
Hilbert’s axioms - Betweenness
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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
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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.
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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.
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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
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Notation
By
A*B*C we will mean “the point B is between the point
A and the point C.”
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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.
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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?)
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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.
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EXERCISE
Let A and B denote two points. Prove that AB ∩ BA = AB AB U BA = AB
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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.
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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?
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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.
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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.
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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.
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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
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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.
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Proposition
If A*B*C then B is the only point lying on the rays BA and BC and AB=AC.
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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.
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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.
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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
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Definition
Ray AD is between rays AC and AB if AB and AC are not opposite rays and D is interior to <CAB.
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Crossbar theorem
If the ray AD is between rays AC and AB then AD intersects segment BC
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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?