Download Logic Rules (2)

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
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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
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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 ~ [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 (1)
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• 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)
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• 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
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• 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
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• 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
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• 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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Incidence Geometry
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• 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
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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
Lines
A
a = {A,C}
B
b = {A,B}
C
c = {B,C}
System 2: 3 lines in the "universe" a,b,c
U = {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 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)
l || m (parallel)
x  y (similar)
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 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].
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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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