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Math 3100
Scribe for 9/16/09
A few clarifications before we begin
When denote something is unique we use an exclamation point.
(For example: ! reads thereexists a uniqueline )
Because can mean length of the segment we will denote a segment
as AB. Left and right does not have meaning in relations to lines,
because lines can be drawn vertically.
To determine where points lie on the line we can reference the
betweenness axioms.
AB = { } is a set with one element therefore it is incorrect to
use this notation. We can only talk about points that are not
incident with the line as being on one side or
another of that line .
When we claim that point Q is an element of then we claim that
is a set although it is not.
B-4 For every line and for any three points A, B, and C not
lying on :
(i) If A and B are on the same side of and B and C are on the
same side of , then A and Care on the same side of .
Explanation: Points A, B and C were perceived to be on the same
side of , which was
represented by a drawing. To confirm that our drawing was
correct we will reference
Definition 2.1, which states that points A and B are on the same
side of because segment
AB does not contain any points on . Similarly, points B and C
are on the same side of
because segment BC does not contain any points on . From the
definition as well as the
axiom, we conclude that points A and C are on the same side of
.
(ii) If A and B are on opposite sides of and B and C are on
opposite sides of , then A and Care on the same side of .
Explanation: Point A was perceived to be on the opposite side of
as points B and C, which
was represented by a drawing. To confirm that our drawing was
correct we will reference to
Definition 2.1, which states that points A and B are on opposite
sides of if the segment AB
does contain a point on . Similarly, points B and C are on
opposite sides of because
segment BC does contain a point on . From the definition as well
as the axiom, we
conclude that points A and C are on the same side of .
Proposition 2.2 For any two distinct points A, B the following
holds:
(i) AB = BABy I-1, for two distinct points A, B as given, there
! line .
By B-2, given two distinct points A and B there exist points C,
D, and E lying on such that
C*A*B and A*D*B and A*B*E.
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By B-1, A, D, and B are distinct points all lying on the same
line such that A*D*B and B*D*A.
By definition of a segment, AB = {D: A*D*B} U {A,B} = {D: B*D*A}
U {B, A} =BA. We can claim
that AB = BA by axiom B-1. Q.E.D.
Note: Segment AB is essentially a collection of points,
therefore another proof of
Proposition 2.2 (i) can be shown by AB BA and BA AB.
(ii) AB (a proper subset)A definition of a ray is necessary to
prove Proposition 2.2 (ii).
Definition of a ray was first proposed to be = {C: A*B*C} U {D:
A*C*B} U {A}. However,
point B was not accounted for in this definition and another
definition was proposed which
included the definition of a segment which accounted for point
B, the following definition is
the definition we will use for a ray.
Definition of a ray: = {C: A*B*C} U AB
Proof: AB by definition of .
To confirm that AB we can confirm that {C: A*B*C} by B-2.
Proposition 2.3 For any two distinct points A, B the following
holds:
(i) = ABExplanation: If we construct ray AB and BA the
intersection of the two rays is what
the two rays have in common which is segment AB.
(ii) U =Explanation: If we construct ray AB and BA the union of
the two rays will be all ofray AB and all of ray BA which is .
Lemma 2.4 Points A and B are on the same side of iff side (A, )
= side (B, ).
Note: side (A, ) can be read as The side A is on in relation to
. To prove the lemma we will
need to prove both possible cases because if and only if (iff)
is explicitly stated.
**Proof will need to be completed for Monday September 21st,
however, a few ideas for the
proof were presented.**
side (A, ) = {P: P and A are on the same side of } side (B, ) =
{P: P and B are on the same side of } Proof in one direction: If A,
B are on the same side of side of then side (A, ) = side (B,).
Proof in the other direction: If side (A, ) = side (B, ) then A and
B are on the same side of .
Proposition 2.5 Every line has exactly two sides and they are
disjoint.
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By B-2, there exist two distinct points A, C and we can
determine that there is also a point B such that
A*B*C. If a line is drawn through point B we can claim that by
Definition 2.1 as well as Lemma 2.4 that
side (A, ) side (C, ).
If we select an additional distinct point Q, if AQ then Q is on
the opposite side of A. Or if AQ does not
intersect , then Q is on the same side of A. By B-4, we can
determine that Q and C are on the same sideof .
Lemma 2.6 If A*B*C and is any line passing through C that is
distinct from line AC, then A and B are
on the same side of .
Given A*B*C, is any line passing through C.
By B-1, A*B*C and C*B*A where A, B and C are three distinct
points lying on the same line .
By Proposition 1.2 line AC and have a unique point, C, in
common.
Definition 2.1 implies that AB does not intersect line therefore
A and B are on the same side of .
