Notes by Michael K. Pellegrino

Class #1 - Tuesday, July 10, 2012

__Incidence Axiom I__ - Given any two disticnt points, P and Q, there exists exactly one line *L* such that P and Q lie on *L*.

__Incidence Axiom II__ - For every line *L* there exists at least two distinct points, P and Q, such that both P and Q lie on *L*.

__Incidence Axiom III__ - There exist three points that do not lie on any one line.

__Colinear__ - Three points A, B, and C are __colinear__ if there exists a line *L* such that all three points lie on *L*

__Non-colinear__ - Three points A, B, and C are __non-colinear__ if there does not exist a line *L* such that all three points lie on *L*

__Fano's Geometry__

Interpret Point to be one of A, B, C, D, E, F, and G

Interpret Line to be one of the seven triples: (ABC), (CDE), (EFA), (AGE), (CGF), (EGB) and (BDF)

Do all three incidence axioms work for this geometry? *yes.*

An *interpretation* of an axiomatic system is a particular/specific way of giving meaning to the undefined terms in the system.

*Interpretation* is a __model__ if the axioms are __all__ true in that interpretation.

Axiom I | Axiom II | Axiom III |
---|---|---|

Given any 2 distinct pts, there exists exactly 1 line | For every line there exists at least 2 distinct pts | There exist 3 pts that do not lie on any 1 line |

True | True | True |

A statement in our axiomatic system is independent of the axioms if it *cannot* be proven or disproven as a logical consequence of the axioms.

If we can make 2 models up, where in one of which, the statement is true and in the other of which the statement is false, then we know the statement is independent of the axioms.

**Definition 2.3.1: Parallel** - Two lines L and M are said to be __parallel__ if they lie in the same plane and there is no point P such that P lies on *both* L and M. Symbolically written: L || M.

**Euclidian Parallel Postulate** - For every line L and for every point P not on L there is *exactly* one line M such that P lies on M and M || L.

**Elliptic Parallel Postulate** - For every line L and for every point P not on L there is *no* line M such that P lies on M.

**Hyperbolic Parallel Postulate** - For every line L and for every point P not on L there are *at least two* lines M and N such that P lies on M and N and M || L and N || L.

**Concerning Spheres**

Interpret __point__ to be an ordered triple of real numbers (x, y, z) with x² + y² + z² = 1

Interpret __line__ to be a *Great Circle* on a sphere. (A Great Circle is a circle whose radius is the radius of the sphere.)

The intersection of the plane that passes through the origin (0, 0, 0) and the two points involved on the sphere with the sphere.

The two points on the sphere are said to be __Antipodal__ (opposite) if they are points on which a line through the origin intersects the sphere.

Two given __antipodal__ points on the sphere lie on an infinite number of __Great Circles__.

Incidence Axiom I is **NOT** satisfied using two antipodal points on a sphere so this is **NOT** a model for incidence geometry.

It *does* however satisfy the __elliptic parallel postulate__.

**Five Point Geometry**

Interpret *points* to be one of A, B, C, D and E

Interpret *line* to mean a set of exactly two points.

Axiom I | Axiom II | Axiom III |
---|---|---|

Given any 2 distinct pts, there exists exactly 1 line | For every line there exists at least 2 distinct pts | There exist 3 pts that do not lie on any 1 line |

True | True | True |

Also, __Hyperbolic Parallel Postulate__ holds because for every line *L* and for every point where P=A,B,C,D or E not on *L*, there are *at least* two lines M and N such that P lies on M and P lies on N.

**Four Point Geometry**

Interpret point to be one of A, B, C or D.

Interpret line to be a set of two points.

Axiom I | Axiom II | Axiom III |
---|---|---|

Given any 2 distinct pts, there exists exactly 1 line | For every line there exists at least 2 distinct pts | There exist 3 pts that do not lie on any 1 line |

True | True | True |

Also, __Euclidean Parallel Postulate__ holds because for every line L and for every point P not on L there is *exactly* one line M such that P lies on M and M || L.

**Three Point Geometry**

Interpret point to be one of A, B or C.

Interpret line to be a set of two points.

Axiom I | Axiom II | Axiom III |
---|---|---|

Given any 2 distinct pts, there exists exactly 1 line | For every line there exists at least 2 distinct pts | There exist 3 pts that do not lie on any 1 line |

True | True | True |

Also, __Elliptic Parallel Postulate__ holds because for every line L and for every point P not on L there is *no* line M such that P lies on M.

**Two Point Geometry**

Interpret point to be one of A or B.

Interpret line to be a set of two points.

Axiom I | Axiom II | Axiom III |
---|---|---|

Given any 2 distinct pts, there exists exactly 1 line | For every line there exists at least 2 distinct pts | There exist 3 pts that do not lie on any 1 line |

True | True | False |

This is not incidence geometry because it doesn't satisfy Axiom III.

**Theorem 2.5.1** - Lines that are **NOT** Parallel intersect in exactly one point in a plane.

**Proof** - Let L and M be two distinct lines in the same plane that are not parallel.

L and M *must* intersect in at least one point P (by the definition of parallel). If L and M *are* parallel then there is no point R such that R lies in *both* L and M.

P → Q

*NEGATION* - if P → Q is *NOT* true then there is a point R such that R is on *both* L and M.

If L and M are not parallel then there is a point R such that R is on both L and M.

**R.A.A.** - Reductio ad absurdium - Proof by contradiction.

Suppose lines L and M intersect in **two** points R and S then the points R and S are both on line L... this violates Incidence Axiom I. This contradiction means that lines L and M cannot intersect in more than one point hence L and M are distinct, non-parallel lines and they *must* intersect in exactly one point.

**Converse** - if Q is true then P is true. Q → P?

If L and M intersect in exactly one point then L and M are distinct and non-parallel.

In this case, the *converse* is indeed true, but this is not always the case.

**Proof**: Suppose L = M then there exists exactly one point R that lies on both L and M (given). Incidence Axiom II says that L must contain **two** points R and S. Since L is supposed to equal M then R and S would have to lie on *both* L and M. This last statement contradicts Incidence Axiom II hence L must *NOT* be equal to M. We are given that L and M intersect so that means that they are not parallel.

**Contrapositive** - if P → Q then !Q → !P is also true.

This is because of the following truth tale:

P | Q | !P | !Q | P → Q | !Q → !P |
---|---|---|---|---|---|

T | T | F | F | T | T |

T | F | F | T | F | F |

F | T | T | F | T | T |

F | F | F | T | T | T |

**Theorem** - if x = 0 then x² = 0

Contrapositive - if x^{2} ≠ 0 then x ≠ 0

Converse - if x^{2} = 0 then x = 0

Contrapositive of the Converse - if x ≠ 0 then x^{2} ≠ 0

**Conditional Statements** - if P is true then Q is true (P → Q). P is the __hypothesis__ and Q is the __conclusion__.

A theorem is a conditional statement that has been proven to be true.

If A ≠ 0 then A^{2} ≠ 0 (a conditional statement)

This is not a theorem for matrices because there is a counter example

Once a theorem has been formed (meaning it's been proved to be true) we can write P __⇒__ Q

Example: If x is in **ℝ** and x^{2} < 0 then x = 4

Note... the conclusion is true for all x for which the hypothesis is true.

There happen to not be any x's for which the hypothesis is true, so there are no counterexamples to this theorem. It is true... but we say that this theorem is *vacuously* true.

It's like saying that all cell phones in the room are turned off when there aren't any cell phones in the room. All of them *ARE* off, but all the cell phones in the room are *also* on because there aren't any cell phones. So one could also say that "All the cell phones in the room are on **and** off" because of the vacuous nature of this truth.

Example: If x is irrational then x^{2} is irrational. This is false because √2 is irrational, but (√2)^{2} = 2 which is rational.

**THEOREM 2.6.3** - If L is any line then there exists at least one point P such that P does not lie on L. (Assume Incidence Geometry).

**PROOF** - Let L be any line then there are three points (A, B and C) that are non-colinear (Incidence Axiom III)

Hence at least one of them doesn't lie on L

**THEOREM 2.6.4** - If P is any point there are at least two distinct lines L and M such that P lies on both L and M

**PROOF** - Let P be any point. By Incidence Axiom III there exist three non-colinear points A, B and C.

**Case 1**: P is one of those three points

P = A

L = Line AB and M = Line AC (by Incidence Axiom I)

P=A lies on both of these lines

Suppose L = M... this is not possible because they'd have to be co-linear so L ≠M and the theorem is true. It follows __W__ithout __L__oss __O__f __G__enerality the same if P=B or P=C

**Case 2**: Suppose P is NOT one of these points.

Let L = line PA, M = line PB and N = line PC (by Incidence Axiom I)

These three lines *cannot* be the same because that would imply that A, B and C are co-linear... hence at least two of the three lines are distinct and P lies on both of those.

Class #2 - Thursday, July 12, 2012

Chapter 3

Cenema's Axioms for Two Dimenstional Plane Geometry

**Axiom 3.1.1 - Existance Postulate** - the set of all points is a non-empty set with at least two points

The set of all points is called the Plane and denoted **P**

A line through the points A and B is denoted by AB with a double-arrowed line over it.

We write A in L to indicate the point A lies on the line L, or a is incident with L

**Axiom 3.1.3 - Incidence Postulate** - Every line is a set of points where every pair of distinct points (A and B) is exactly one line L on AB such that A and B are incident with L. A, B in L

A point Q is extetrnal to L if Q is not incident with L

Two lines L and M are parallel if they lie in the same plane *and* there is no point P with P on L and P on M. L || M or L AND M = NULL

**Theorem 3.1.7**: if L and M are two distinct non-parallel lines then there exists exactly one point P that lies on both L and M.

**Corollary** - If L and M are lines, the three possibilities are that: L=M, L||M, or L AND M contains exactly one point.

**Axiom 3.2.1 - Ruler Postulate** - for every pair of points (P and Q) there exists a real number **ℝ**. PQ ≡ ||PQ|| is the distance from P to Q and there exists a line L. There is a 1:1 mapping

F:l → **ℝ** such that if P and Q are points on the line corresponding to the x,y in **ℝ**, x = f(P) and y = f(Q), then we're claiming that the distance from P to Q is PQ=||x-y||.

Let L be a line, a 1:1 correspondence (meaning a function that is 1:1 and onto) f:l → **ℝ** such that PQ = |f(P) - f(Q)| for all points PQ in L is called a __coordinate function__ for L and the number f(P) is called the coordinate of P.

Def: Let A,B,C be distinct points. We say that B is between A and C and write A*B*C if A, B, and C are colinear. C ∈ AB and AB + BC = AC. Also f(A) < f(B) < f(C) or f(C) < f(B) < f(A). It turns out that if AB + BC = AC for three distinct points in a plane A, B, and C need to be colinear. In general AB + BC __>__ AC (The **Triangle Inequality**).

The segment joining A and B in AB: AB = { A, B } U { P | A*P*B }. A in f(A), B in f(B) with A < B. [A,B] = { x | A __<__ X __<__ B is AB^{>} is the ray from A in the direction of B. AB U { P | A * B * P }

Def: the Length of AB denoted by AM is the distance from A to B.

We call A and B the endpoints of AB and two segments AB and CD are congruent if AB = CD