Math 3100
Scribe for 9/16/09
A few clarifications before we begin…
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When denote something is unique we use an exclamation point. (For example: ∃! ℓ reads “there
exists a unique line ℓ)
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 ℓ as being on one side or
another of that line ℓ.
When we claim that point Q is an element of ℓ then we claim that ℓ is a set although it is not.
B-4 For every line ℓ and for any three points A, B, and C not lying on ℓ:
(i)
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.
Explanation: Points A, B and C were perceived to be on the same side of ℓ, 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 ℓ because segment
AB does not contain any points on ℓ. Similarly, points B and C are on the same side of ℓ
because segment BC does not contain any points on ℓ. From the definition as well as the
axiom, we conclude that points A and C are on the same side of ℓ.
(ii)
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.
Explanation: Point A was perceived to be on the opposite side of ℓ 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 ℓ if the segment AB
does contain a point on ℓ. Similarly, points B and C are on opposite sides of ℓ because
segment BC does contain a point on ℓ. From the definition as well as the axiom, we
conclude that points A and C are on the same side of ℓ.
Proposition 2.2 For any two distinct points A, B the following holds:
(i)
AB = BA
By 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
C*A*B and A*D*B and A*B*E.
such that
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:
Proof: AB
= {C: A*B*C} U 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)
∩
= AB
Explanation: 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 of
ray AB and all of ray BA which is
.
Lemma 2.4 Points A and B are on the same side of ℓ iff side (A, ℓ) = side (B, ℓ).
Note: side (A, ℓ) can be read as “The side A is on in relation to ℓ”. 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.**
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side (A, ℓ) = {P: P and A are on the same side of ℓ}
side (B, ℓ) = {P: P and B are on the same side of ℓ}
Proof in one direction: If A, B are on the same side of side of ℓ then side (A, ℓ) = side (B, ℓ).
Proof in the other direction: If side (A, ℓ) = side (B, ℓ) then A and B are on the same side of ℓ.
Proposition 2.5 Every line has exactly two sides and they are disjoint.
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, ℓ) ≠ side (C, ℓ).
If we select an additional distinct point Q, if AQ ∩ ℓ then Q is on the opposite side of A. Or if AQ does not
intersect ℓ, then Q is on the same side of A. By B-4, we can determine that Q and C are on the same side
of ℓ.
Lemma 2.6 If A*B*C and ℓ is any line passing through C that is distinct from line AC, then A and B are
on the same side of ℓ.
Given A*B*C, ℓ 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 ℓ have a unique point, C, in common.
Definition 2.1 implies that AB does not intersect line ℓ therefore A and B are on the same side of ℓ.