Ch2
Chapter 2: Logic & Incidence Geometry
Back To the Very Basic Fundamentals
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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.
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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)
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 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 same thing as [~ S
1
~S
2
1
S
2
] means the
]. (
means “or”)
•
A contradiction (absurd statement) is a statement of the form S
~S. (
)
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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).
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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).
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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!)
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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)
•
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 already a valid step in a proof. Suppose that is 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.
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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 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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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.
System 1: 3 points in the "universe" A, B, C
U = { A, B, C}
Points
A
Lines a = {A,C}
B
C b = {A,B} c = {B,C}
System 2: 3 lines in the "universe" a,b,c
U = {a,b,c}
Lines Points
a A = {a,b}
b
c
B = {b,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 a t 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 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)
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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 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 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}.
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