Question

Consider the following axiomatic system: 1. Each point is
contained by precisely two lines. 2. Each line is a set of four
points. 3. Two distinct lines that intersect do so in exactly one
point.

True or false? There exists a model with no parallel lines.

Answer #1

Consider an axiomatic system that consists of elements in a set
S and a set P of pairings of elements (a, b) that satisfy the
following axioms:
A1 If (a, b) is in P, then (b, a) is not in P.
A2 If (a, b) is in P and (b, c) is in P, then (a, c) is in
P.
Given two models of the system, answer the questions below.
M1: S= {1, 2, 3, 4}, P= {(1, 2), (2,...

3. Consider the following two lines:
x = c + t, y = 1 + t, z = 5 + t and x = t, y = 1 - t, z = 3 +
t.
Is there a value c that makes the two lines intersect? If so,
find it. Otherwise, give a reason.
4. A particle starts at the origin and moves along the shortest
path to the line determined by the two points P =(1,2,3) and Q
=(3,-2,-1)....

Consider the lines in space whose parametric equations are as
follows
line #1 x=2+3t, y=3-t, z=2t
line #2 x=6-4s, y=2+s, z=s-1
a Find the point where the lines intersect.
b Compute the angle formed between the two lines.
c Compute the equation for the plane that contains these two
lines

Given the following Axioms of four-point geometry, prove whether
or not Axiom 3 is independent of the other Axioms:
1. There are exactly four points
2. Each pair of points are on exactly one line.
3. Each line is on exactly two points.

A. Explain the connections between geometric constructions and
two of Euclid’s postulates.
1. A straight line segment can be drawn joining any two
points.
2. Any straight line segment can be extended indefinitely in a
straight line.
3. Given any straight line segment, a circle can be drawn having
the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way
that the...

2.For each of the following, give a concrete example. Explain in
max. 3 lines why your example has the stated property.
(c) An equivalence relation on N that has exactly three
equivalence classes.
(d) An ordering relation on the set {a, b, c, d} that does not
have a maximum element.
1. [10 points] For each of the following statements, indicate
whether it is true or false. You don’t have to justify your
answers.
(i) If R is an equivalence...

Refer the axioms below
Ques
For each part of the five axioms, decide whether it is
true or false for a cylinder. Explain how you know that it is true
and/or exactly when and how it fails. Make sure that you consider
each possible type of line and every possible configuration of
points. Write down the best possible variation of the axiom which
is true. Draw pictures where appropriate.
Note: To test the plane separation axiom, try actually cutting the...

1. This exercise asks you to
negate various results that are equivalent to the Euclidean
parallel postulate.
(a) Alternate Postulate 5.1. Given a line and a
point not on the line, exactly one line can be drawn through the
given point and parallel to the given line.
(b) Alternate Postulate 5.2. If two parallel lines
are cut but a transversal, then the alternate interior angles are
equal, each exterior angle is equal to the opposite interior angle,
and sum of...

3. Which of the following sets spans P2(R)?
(a) {1 + x, 2 + 2x 2}
(b) {2, 1 + x + x 2 , 3 + 2x + 2x 2}
(c) {1 + x, 1 + x 2 , x + x 2 , 1 + x + x 2}
4. Consider the vector space W = {(a, b) ∈ R 2 | b > 0} with
defined by (a, b) ⊕ (c, d) = (ad + bc, bd)...

1-Power Systems: Line Diagrams
(a) Sketch a single line diagram for the following power system.
There are three 3-phase alternators supplying an 11 kV busbar and
one 3-phase alternator supplying a 33 kV busbar. There are two
parallel transmission lines fed from the 11 kV line and supplying a
33kV busbar. Each has a step-up and step-down transformer. One
secondary feeders from the 33 kV bus bar feeds a further kV busbar
through a stepdown transformer. This is further stepped...

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