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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

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 ?

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

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.

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.

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.

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.

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.

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.

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.

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

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.

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 .

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.

- 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?

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

Now try Exercise set 1.

1,2,3,4

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.

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

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

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 .

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.

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)

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 )

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.

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.

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