Lesson 1:

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Lesson 1   Axiomatic systems and Incidence
Geometry Outline Axiomatic systems
Models of axiom systems   An example of Axiom
system   Parallel postulates   Conditional
propositions and proof
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Example Let D and E be the midpoints of the side
AB and AC of
respectively. Then DE BC.

• Questions
• What does this sentence mean?
• What is the midpoint?
• What does mean?
• What is DE B ?
• Is the statement correct? Why ?

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1.1 . Axiomatic system 1.1.1. Example. In
Euclidean geometries, we have the following
Point, lines, perpendicular lines, triangle,
angle, rectangle, similar triangle,   It is
possible to draw a straight line through any two
given points It is possible to
construct a circle with any given center and
radius Any two right angles are
equal
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1.1.2. Main components of an axiomatic system
Undefined and Defined terms Undefined terms are
the technical words that will be used in the
subject. Defined terms are the words that are
defined or described using undefined terms and
those have been defined previously.
Axioms ( postulations) The axioms are
statements that are accepted without proof.  The
axioms are given meaning to undefined terms.
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Theorems ( propositions) Theorems are the
statements that can be proved from the
axioms or proven theorems. Models An
interpretation of an axiom system is a particular
way of giving meaning to the undefined
terms in that system. A model of an axiom
system is an interpretation of the system such
that all the axioms are corrected statements in
that interpretation.
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1.1.3. Example Euclidean geometry, Undefined
terms point and straight line are undefined
terms. Postulations 1). Any two points can be
joined by a straight line. 2). Any straight
line segment can be extended indefinitely in a
straight line. 3). Given any
straight line segment, a circle can be drawn
having the segment as radius and one
endpoint as center. 4). All right angles
are congruent. 5). If two lines are drawn
which intersect a third in such a way that the
sum of the inner angles on one side is less
than two right angles, then the two lines
inevitably must intersect each other on that
side if extended far enough.
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The Postulation 5) is equivalent to the
parallel postulation in this geometry  
Through a point not on a given straight line, one
and only one line can be drawn that never meets
the given line.  
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1.1.4. Independent statements, consistent system
A statement is independent of the axioms if it
not possible to prove or disprove it from the
axioms. A good way to prove the independence
of a statement is to find one model in which the
statement is true and another model in which the
statement is false. An axiom system is called
consistent if there are no two contradict
theorems in the system. If there is a model
for the system, then it is consistent.
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1.2 . An example of an axiomatic system(
Incidence geometry ) Undefined terms
point, line, lie on (incident) Definition
Three points A, B and C are collinear if there
exists one line l such that three of the points
A, B and C all lie on l. The points are
noncollinear if there is no such line l.
Axioms (IA 1) For every pair of points P and Q
there exists exactly one line l such that
both P and Q lie on l. (IA 2) For every line l
there exist at least two distinct points P and
Q such that both P and Q lie on l. (IA 3)
There exist three noncollinear points.
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1.2.1. Example The three-point plane model.
Interpretation a point means one of
the symbols A, B and C a line is one of
the sets A, B, A,C or B,C. lie on
means is an element of
Exercise Check that all the axioms of
incidence geometry are satisfied.
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1.2.2.Example Points A, B,C Line A,
B, C Which axioms are satisfied?
1.2.3. Example Fanos geometry Points A, B,
C, D, E, F, G Lines A,B,C, C,D,E,
E,F,A, A,G,D, C,G,F,
E,G,B, B,D,F
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1.2.4. Example The Cartesian plane A point is
a pair (x,y), x,y are real numbers A line is a
set of points whose coordinates satisfy a linear
equation of the form axbyc0, where a,b and c
are fixed real numbers for the line and a and b
are not both 0. A point is said to lie on a line
if the coordinates of the point satisfy the
equation.
1.3. Parallel postulates 1.3.1.Definition Two
lines l and m are said to be parallel if there is
no point P such that P lies on both l and m.
If l and m are parallel, we write l m.  
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1.3.2. Three Parallel postulates  Euclidean
Parallel Postulate For every line l and for
every point P that does not lie on l, there is
exactly one line m such that P lies on m and l
m.
Elliptic Parallel Postulate For every line l
and for every point P that does not lie on l,
there is no line m such that P lies on m and
l m.
Hyperbolic Parallel Postulate For every line l
and for every point P that does not lie on l,
there are at least two lines m and n such
that P lies on both m and n and m and n are
both parallel to l .
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1.3.3. Example (a) The three-point plane
does not satisfy Euclideans Parallel Postulate.
It satisfies the Elliptic Parallel postulate.
(b)    The four-point geometry satisfies
the Euclidean Parallel postulate. Points
A,B,C,D Lines A,B,A,C,A,D,B,C,B,D,C
,D (c)
The Cartesian plane satisfies the Euclidean
Parallel Postulate.
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• Exercise Consider the five-point model.
• Points A,B,C,D,E
• Lines All pairs of points
• Lie on means is a member of  
• Which Parallel Postulates does the five-point
  model satisfy?

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The parallel postulates is independent of the
axioms on incidence geometry.
Summary 1. An axiom system is determined by
some undefined terms, axioms . 2. A theorem is a
statement that can be proved from the axioms or
other theorems 3. A model of a system is an
interpretation of the system so that all the
axioms of the system are corrected statements
in that interpretation. 4. Incidence geometry 5.
There are three different Parallel postulations
Euclidean Parallel Postulate Elliptic
Parallel Postulate Hyperbolic Parallel
Postulate
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Now try Exercise set 1.
1,2,3,4
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1.4. Conditional propositions 1.4.1. Propositions
A proposition in mathematics means a
statement which is either true or false but not
both. (1)   The addition of two even numbers is
an even number.  (2) x2lt5. (3) If ylt5,
then y2lt7.  (4) Goldbach conjecture If mgt2
is an even integer, then there are prime numbers
p and q such that mpq. (5) If xlt 5 and ylt7,
then xylt20.
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1.4.2. Conditional propositions If l is a
line, then there exists at least one point P
such that P does not lie on l.  
Hypothesis Conclusion
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1.4. 3. Example A father told his son If you
get an A in your next math exam, I will buy a
new lap-top for you. In which case(s) do you
think the father breaks his promise?
1.5. Inverse, converse and contrapositive  For
any given proposition p, the negation
of p is the proposition which is true only
when p is false.
From the proposition , we
can form three new propositions Converse
Inverse Contrapositive
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Every conditional proposition is equivalent to
its contrapositive.
1.5.1. Example Let m and l be two distinct
lines. If m and l are parallel, then there
exists no point that lies on both m and
l. Converse If there exists no point that lie
on both m and l, then m and l are
parallel. Inverse If m and l are not parallel,
then there is a point that lie on both m and
l. Contrapositive If there exists a point
that lie on both m and l, then m and l are not
parallel .
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Exercise State the inverse, converse and
contrapositive of the following proposition
If ABCD is a parallelogram, then the two
diagonals AC and BD bisect each other.
1.6. Proof of a proposition 1.6.1.Example If l
is a line, then there exists a point P such that
P does not lie on l. Proof 1. l is a line

( Assumption) 2. There exist three
distinct points P , Q and R that are not
collinear
( By Axiom 3) 3. At least one of P, Q and
R does not lie on l ( By 2) 4. There is a
point that does not lie on l.
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A proof of a proposition consists of a
sequence of statements each of them is either an
assumption , or an axiom, or a statement
derived from the previous statements. The last
statement is the conclusion.

• Proof
• Statement ( reason)
• Statement (reason)
• m. Conclusion (reason)

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1.6.2. Proof by contrapositive Since the
proposition If P then Q is
equivalent to its contrapositive , we can prove
If P then Q by proving its contrapositive
If not Q then not P.

• 1.6.3. Example If m and n are integers and mn
  is an even number, then at least one of m and n
  is even.
• Proof
• Suppose both m and n are odd numbers (
  Contrpositive
• 
  assumption)
• 2. m2p1, n2q1 for some integers p and q (
  by 1 )
• 3. mn(2p1)(2q1)
• 2(2pqqp)1
  ( By 2 )
• 4. mn is odd
  ( By 3 )

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Exercise Prove the following by
contrapositive If x and y are two integers whose
product is odd, then both x and y must be odd.
1.7. Indirect proof ( Proof by contradiction) To
prove If P then Q by contradiction we assume
P is true and Q is not true and derive two
contradict propositions.
To prove If P then Q by contradiction we
assume P is true and Q is not true and derive two
contradict propositions.
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Example If m and l are distinct , nonparallel
lines, then there exists a unique point that lies
on both m and l.

• Proof
• m and l are distinct lines, m and l are not
  parallel.
• 
  (Assumption)
• 2. there exists a point P lie both on m and l
  ( by 1 )
• 3. suppose there is another point Q that also lie
  on
• both m and l.
• m and l are the same.
  ( By 2, 3 and
• 
  Axiom1)
• 5. so P is the only point that lies on both m and
  l.

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• Thank You!
• Submit the solutions of Exercise Set 1
• Q4, Q5 , Q 8
• by 22 Jan 2008