Chapter 2 Logic Incidence Geometry Back To the

Chapter 2: Logic & Incidence Geometry Back To the Very Basic Fundamentals 1 Copyright, 1996 © Dale Carnegie & Associates, Inc.

Theorems 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 o statements, along with a justification for each statement, ending with the conclusion expected. 2

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 3

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: ) 4

Logic Rules(Some of De. Morgan’s Laws) (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 ~ [S 1 S 2] means the same thing as [~ S 1 ~S 2]. ( means “or”) • A contradiction (absurd statement) is a statement of the form S ~S. ( ) 5

Logic Rules: Quantifiers (1) (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). 6

Logic Rules: Quantifiers (2) (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). 7

Logic Rules: Implication (6) • Rule 8: If P Q and P are several steps in a proof, the 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!) 8

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) 9

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 S 1 S 2 … Sn is already a valid step in a proof. Suppose that the proofs of C are carried out from each of the case assumptions S 1, S 2 … Sn. Then C can be concluded as a valid step in the proof. 1 0

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. 1 1

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. 1 2

Example 5: Isomorphism -- 1) one and only one element goes to each member of the other set. 2) All elements in the range are used up. 1 3

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 a t least three distinct points lying on it. An affine plane is a model of incidence geometry having the Euclidean parallel property 1 4

Equivalence Relations • An equivalence relationship ”~” between two objects “a and b” is a relationship with these three properties: 1. a ~ a, i. e. a is equivalent to itself. (reflexive) 2. a ~ b b ~ a. (symmetric) 3. [a ~ b b ~ c] [a ~ c]. (transitive). • Examples of equivalence relations: a = b (equality) l || m (parallel) x y (similar) p q (perpendicular) • Example of relations not equivalence classes. G < H (less than) C D (proper subset) 1 5

Equivalence 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 Aand 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]. 1 6

Points 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}. 1 7