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Copyright, 1996 © Dale Carnegie & Associates, Inc.

Chapter 2: Logic & Incidence GeometryChapter 2: Logic & Incidence Geometry

Back To the Very Basic Fundamentals

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Logic Rule 0Logic Rule 0

• No unstated assumptions may be used in a proof!

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Theorems and ProofsTheorems and Proofs

• A mathematical theorem is a conditional statement of the form:

If H, then C. (In symbols: H C)

• A mathematical proof is a list of statements, along with a justification for each statement, ending with the conclusion expected.

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Logic Rules (1)Logic Rules (1)

Rule 1:The following are the six types of justifications allowed for statements in proofs:1. By hypothesis. . .

2. By axiom . . .

3. By theorem . . .

4. By definition . . .

5. By (previous) step . . .

6. By rule . . . of logic

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Logic Rules (2) Logic Rules (2)

Rule 2: Indirect Proof [redutio ad absurdum (RAA)] :

• To prove a statement H C, assume the negation of statement C (RAA hypothesis and deduce an absurd statemtent, using H if needed.

• To prove: H C

1. Assume H ~C (Symbol for negation of C: ~C)

2. Use this idea to arrive at a contradiction to H or some other known theorem, definition or axiom. ( Symbol for contradiction: )

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Logic RulesLogic Rules(Some of DeMorgan’s Laws)(Some of DeMorgan’s Laws) (3) (3)

• Rule 3: The statement ~(~S) means S.

• Rule 4: The statement ~[H C] is the same statement as H & ~C. (& and mean “and”)

(Alternate symbols: H ~C)

• Rule 5: The statement ~ [S1 S2] means the same thing as [~ S1 ~S2]. ( means “or”)

• A contradiction (absurd statement) is a statement of the form S ~S. ()

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Logic Rules: Quantifiers Logic Rules: Quantifiers (1)(1) (4) (4)

• Quantifiers are of two types:– Universal: For all x …, For any x …, For every x…, If

x is any… (Symbol: x)

(Note: For all… x does NOT imply the existence of anything!)

– Existential: There exists an x…, For some x…, There are x…, There is an x… (Symbol: x)

• Statements involving quantifiers: If S is a statement that says something about x, written S(x), and it is quantified, we write for example: x S(x) or x S(x).

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Logic Rules: QuantifiersLogic Rules: Quantifiers (2) (2) (5) (5)

• Rule 6: The statement ~[x S(x) ] means the same as x ~S(x).

• Rule 7: The statement ~[x S(x)] means the same as x ~S(x).

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Logic Rules: Implication (6)Logic Rules: Implication (6)

• Rule 8: If P Q and P are several steps in a proof, then Q is a justifiable step.

• Conditional Statement: P Q (If P, then Q.)– Its converse: Q P– Its inverse: P ~Q (negation)– Its contrapositive: ~Q ~P

• Logically equivalent: P Q. “P if and only if Q” P is logically equivalent to Q. (P and Q are the same thing!)

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Logic Rules: Tautologies (6)Logic Rules: Tautologies (6)

• Rule 9: Statements that are true strictly because of their form and not what individual parts might “say”.A) [ [P Q ] [Q R] ] [P R] (Transitive)

B) [P Q] P, or, [P Q] Q (Inclusive)

C) [~Q ~P] [P Q] (Contrapositive)

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Logic Rules (7)Logic Rules (7)

• Rule 10: (The Excluded Middle) For every statement P, P ~P is a valid step in a proof.

• Rule 11: (Proof by cases) Suppose the disjunction of statements S1 S2 … Sn is already a valid step in a proof. Suppose that the proofs of C are carried out from each of the case assumptions S1, S2 … Sn. Then C can be concluded as a valid step in the proof.

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Equivalence Relations: Logic Equivalence Relations: Logic Rule 12Rule 12

• An equivalence relationship ”=” between two objects “X and Y” is a relationship with these three properties:1. X (X=X), i.e. X is equivalent to itself. (reflexive)2. X Y (X=Y Y=X). (symmetric)3. X Y Z[(X=Y & Y=Z) X=Z]. (transitive). Also: If X=Y and S(X) is a statement about X, then S(X) S(Y).

• Examples of equivalence relations:a = b (equality) x y (similar) AB CD (congruent)l || m (parallel) p q (perpendicular)

• Example of relations not equivalence classes.c < d (less than) C D (proper subset)

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Incidence Geometry (1)Incidence Geometry (1)

• Incidence Axioms

I-1: For every point P and for every point Q not equal to P there exists a unique line l incident with P and Q.

I-2: For every line l there exist at least two distinct points that are incident with l.

I-3: There exist three distince points with the property that no line is incident with all three of them.

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Incidence Geometry (2)Incidence Geometry (2)Incidence Propositions

P-2.1: If l and m are distinct lines that are not parallel, then l and m have a unique point in common.

P-2.2: There exist three distinct lines that are not concurrent.

P-2.3: For every line there is at least one point not lying on it.

P-2.4: For every point there is at least one line not passing through it.

P-2.5: For every point P there exist at least two lines through P.

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ParallelParallel

• Def of Parallel: l || m if l ~ I m

• Parallel Postulate (Euclid):

l P, ~(P I l) !m (P I m & l || m)

• Notation: P a point, l and m lines, ~ I not incident

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Example 5: Isomorphism -- 1) one and only one Example 5: Isomorphism -- 1) one and only one element goes to each member of the other set. 2) All element goes to each member of the other set. 2) All elements in the range are used up.elements in the range are used up.

System 1: 3 points in the "universe" A, B, CU = { A, B, C}Points Lines A a = {A,C}

B b = {A,B}

C c = {B,C}

System 2: 3 lines in the "universe" a,b,cU = {a,b,c}Lines Points

a A = {a,b}

b B = {b,c}

c C = {a,c}

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Projective and Affine Planes

A projective plane is a model of the incidence axioms having the elliptical property (any two lines meet) and such that every line has at least three distinct points lying on it.

An affine plane is a model of incidence geometry having the Euclidean parallel property

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Equivalence ClassesEquivalence Classes

• An equivalence class C is the set of all objects y equivalent to some object x.

C ={ y : y~x}• Example: Given the affine plane A and a line

l in A (l A) the set of all lines m parallel to l would be an equivalence class and represented by

[l] = {x : x || l, l A} m [l] (m is one of the

x’s. We write m ~ l and also [m] ~ [l].

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Points at InfinityPoints at Infinity

• Points at infinity, by definition, are these equivalence classes defined in the above example.

• The line at infinity l is the set of all the points at infinity! l ={[t] : [t] ~ [l], l any line in A}, i.e. l = {[l], [k],[r] . . . where l, k, r A but none are parallel to each other}.