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

2 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.

3 Logic Rules (1) Rule 1:The following are the six types of justifications allowed for statements in proofs: 1. By hypothesis By axiom By theorem By definition By (previous) step By rule... of logic

4 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:  )

5 Logic Rules (Some of DeMorgan’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. (  )

6 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). …

7 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).

8 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!)

9 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)

10 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  …  S n 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 … S n. Then C can be concluded as a valid step in the proof.

11 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.

12 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.

13 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.

14 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

15 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) x  y (similar) l || m (parallel) p  q (perpendicular) Example of relations not equivalence classes. G < H (less than) C  D (proper subset)

16 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].

17 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}.