MAT 360 Lecture 3
Models and Incidence geometry
Homework due next Tuesday
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Translate Prop. 2.2 and 2.3 to “Shash
geometry” and prove them (in “Shash
words”
Exercise 6 (Only Prop 2.1, 2.4 and 2.5)
Exercises 7 and 8.
Midterm
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Oct 25th
After that we will have one or two
classes in the Math Sync site with the
Geometer’s Sketchpad.
Incidence Geometry:
Undefined terms
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point
line
incident
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Incidence Geometry: Axioms
1. For each point P and for each point Q
not equal to P there exists a unique
line incident with P and Q.
2. For every line T there exist at least
two distinct points incident with T.
3. There exist three distinct points with
the property that no line is incident
with all the three of them.
Two Propositions of Incident
Geometry
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If T and R are distinct lines that there
are not parallel exists a unique point
incident with both of them.
There exist three distinct lines that are
not incident with the same point.
Shash Geometry: Undefined
terms
1. Kem
2. Tam
3. Shash
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Shash Geometry: Axioms
1. For each kem P and for each kem Q
not equal to P there exists a unique
tam shash with P and Q.
2. For every tam T there exist at least
two distinct kems shash with T.
3. There exist three distinct kems with
the property that no tam is shash with
all the three of them.
Definition
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Two tams are squeezed if there is not
kem shashing with both of them
Proposition 1: Dare to prove
it!
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Let T and R are distinct tams that are
not squeezed. Then there is a unique
kem shashing with both of them.
Proposition 2
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There exist three distinct tams that are
not shashed with the same kem.
Definition
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An interpretation of an given axiomatic
system is a “dictionary” that assigns to
each of the undefined terms of the
axiomatic system a particular meaning.
EXAMPLE
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We can interpret the axioms of Shash
geometry as
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Kem = cats
Tam = dogs
Shash = are friends
Friendly Pets Geometry:
Axioms
1. For each cat P and for each cat Q not
equal to P there exists a unique dog
friend with P and Q.
2. For every dog T there exist at least
two distinct cats friend with T.
3. There exist three distinct cats with the
property that no dog is friend with all
the three of them.
Exercise
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Translate Proposition 1 of Shash
geometry to Friendly Pets geometry.
EXAMPLE
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We can interpret the axioms of Shash
geometry as
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Kem = line
Tam = segment
Shash = intersects
Let’s call this interpretation “strange
geometry”.
EXERCISE
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Study the shash axioms with this
interpretation. Are they correct
statements?
Can you use the answer you gave in
the previous question to study the
Propositions of the Shash Geometry
proved ?
Strange Geometry: Axioms
“Interpreted”
1. For each line P and for each line Q not
equal to P there exists a unique segment
intersecting with P and Q.
2. For every segment T there exist at least
two distinct lines intersecting with T.
3. There exist three distinct lines with the
property that no segment intersects all the
three of them.
EXAMPLE
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We can interpret the axioms of Shash
geometry as
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Kem = point
Tam = line
Shash = incidence
Squeezed = parallel
NOTE: We are using the “concrete”
meaning of point, line and incidence here.
EXERCISE
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Study the shash axioms with this
interpretation. Are they correct
statements?
Can you use the answer you gave in
the previous question to study the
Propositions of the Shash Geometry
proved ?
Definition
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An interpretation of the undefined
terms of an axiomatic system for which
the interpreted axioms become correct
statements is called a model.
Three interpretation of Shash
geometry. Which ones are
models?
Kem = line
Tam = segment
Shash = intersects
A small change of direction
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Now that we have played enough with
shash geometry, we are going back to
the usual formulation. So from now
on, our undefined terms are
points, lines and incidence (but
keep in mind that we could be calling
them kems, tams and shash)
Three letter interpretation
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Consider the set {A,B,C}
Points: A, B and C
Lines: {A,B},{B,C},{C,A}
Incidence: “it is a member of” or “contains”
Is this interpretation a model of Incidence
Geometry Axioms?
“Usual plane” interpretation
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Consider a plane P
Points: “usual” points on P
Lines: “usual” lines on P
Incidence: “it is a member of” or “contains”
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Is this interpretation a model?
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Observations: Given an
axiomatic system.
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All the statements deduced from the
axioms are correct statements of a
model.
If there exists a statement that does
not hold in one model, then the
statement cannot follow from the
axioms.
EXERCISE
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Consider Euclid fifth postulate,
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“For every line l and every point P not in l
there exist a unique line through P parallel
to l.”
Is it a correct statement in the “three
letter” model of the Incidence Axioms?
Is it a correct statement in the “usual
plane” model of the Incidence Axioms?
Definition: Given an axiomatic
system,
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A statement (in the language of the
system) is independent of the if
neither the statement or its negation
can be proved from the axioms.
Observation,
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Given an axiomatic system
A statement that holds in one model
and does not hold in another model is
independent.
EXERCISE
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1. Justify the previous observation.
2. Can you give an example of an
independent statement of an axiom
system?
Question
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When are two models isomorphic?
(and what do we mean by
“isomorphic”)
Definition
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Two models are isomorphic if there
exists a one-to-one correspondence
between the interpretation of each set
of undefined terms, such that any
relationship between the objects
corresponding to undefined terms in
one model is preserved, under the
correspondence, in the second model.
Example: Construct an
isomorphism between the models
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Points: A, B and C
Lines:
{A,B},{B,C},{C,A}
Incidence: set
membership
Points: {a,b},{a,c},
{b,c}
Lines: a, b, c
Incidence: set
membership.
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An axiom system is called
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complete if there are not independent
statements.
consistent if it is impossible to deduce
from the axioms a contradiction.
categorical if all the models for the
system are isomorphic.
Exercise: Is the Incidence
Axiom System
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complete?
consistent?
categorical?
AXIOMATIC Interpretation
METHOD
Undefined terms
Axioms
Logic rules
New definitions
Model
Find a model of the following
axiom system
1. For each point P and for each point Q
not equal to P there exists a unique
line incident with P and Q.
2. For every line T there exist at least
two distinct points incident with T.
3. There exist three distinct points with
the property that no line is incident
with all the three of them.
4. There exist exactly three points.
Is the previous axiom
system...
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complete?
categorical?
consistent?
38
Find a model of the following
axiom system
1. For each point P and for each point Q
not equal to P there exists a unique
line incident with P and Q.
2. For every line T there exist at least
two distinct points incident with T.
3. There exist three distinct points with
the property that no line is incident
with all the three of them.
4. There exist exactly four points.
Is the previous axiom
system...
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
complete?
categorical?
consistent?
40