Homework 2 Solutions
April 15, 2015
1. (a) The negation of P Q is ( P )&( Q).
(b) The negation of P &( Q) is ( P ) Q.
(c)
i. By logic rule (3), [ [P = Q] is the same as [P = Q].
ii. By logic rule (4), [P = Q] is the same as [P & Q], so logic rule (9c) imp
Homework 6 Solutions
May 24, 2015
3. Here is a proof of the SAA criterion (Proposition 4.1) that is valid in neutral geometry. Justify
each step.
(a) Assume side AB is not congruent to side DE.
No justication is needed here since it is just an assumption
Homework 4 Solutions
May 6, 2015
1a. Given A B C and A C D. Prove that A, B, C, and D are four distinct points (the proof
requires an axiom).
Proof. By denition A = B, C, D, C = B, D, so all that remains to show is that B = D.
Suppose for a contradiction
MAT 141
Midterm II
Name:
No notes. No calculators. All incidents of cheating or
the appearance thereof will be brought to the attention of
Student Judicial Affairs.
1)
2)
3)
4)
5)
Total (out of 20):
1
1) (5 Points) State the 4 betweenness and the 6 congru
1. Prove that the following statements are equivalent:
P Q and ( P Q) ( Q P ).
2. Write a denial for (!x)A(x), where A(x) is an open sentence.
3. Prove that for every natural number n N, n is an irrational number unless it is
a perfect square, i.e. unless
MAT 141
Midterm 11
Name: KEY
No notes. No calculators. All incidents of cheating or
the appearance thereof will be brought to the attention of
Student Judicial Affairs.
1
[O
)
)_*_
3)_
4)
)
Total (out of 20): 1) (5 Points) State the 4 betweenness and the
Math 432
EXAM 2
April 5, 2013
Name: _
Show your work in each problem, including all major steps. While writing detailed proofs give a justification of each step.
Chapter 3
Finite dierence methods for
two-point boundary value problems
Let us consider a model problem
u (x) = f (x),
0 < x < 1,
u(0) = ua ,
u(1) = ub ,
to illustrate the general procedure using a nite dierence method as follows. Note that,
if f (x) is c
MAT 141
Homework 1 Solutions
The copyright for this text rests with the author.
Problem 3
a: Either line a contains no points or line b contains no points.
b: Line a does not meet line b.
c: Sovanna and Mariela are not related.
d: There is a pair of lines
MAT 141
Homework 2 Solutions
The copyright for this text rests with the author.
Problem 2
a: Michael does well on the quiz but Viktor does not do well.
b: Lina gives a presentation on projective geometry but either Patrick or Richie is not thrilled.
c: Ji
MAT 141
Homework 8 solutions
The copyright for this text rests with the author.
1) Logic Assume that the following statements are true.
David, Elizabeth and Huy are on the soccer team.
Kim, Mimi and Parker are on the golf team.
Stewart and Samir are on th
Definitions for Midterm 2:
Segment - The segment AB is the set of all points between A and B together with the endpoints
A and B.
Ray - The ray AB is the set of all points on the segment AB together with all the points C such
that A B C.
Ordered Incidence
MAT 141
Sample Final
1) (4 Points) True or False:
a: Hilberts axiom of parallelism is the same as the Euclidean parallel postulate.
b: One of the congruence axioms is the side-angle-side
(SAS) criterion for congruence of triangles.
c: Euclidean geometry i
Problem from MAT 141 Discussion (5/3)
Here is the problem that I posed in section this week, together with a solution. Recall that an ordered
incidence plane is an interpretation of incidence geometry which also satisfies the betweenness axioms. In
additi
() By proposition 3.4, every point P on the line incident to A, B, C is either on the ray BC or on the
opposite ray BA. Also, each point on the segment AC is on at least one of these rays.
Let P be in AC. By definition of a segment, either P = A, P = C, o
AFFINE SPACES
1. Definitions
Suppose that V is a vector space. It defines the affine space A whose points are vectors
in V . Lines in A are sets of the form
L = v + Ru = cfw_v + tu : t R,
for various points v and nonzero vectors u. Note that such line L p
Homework #9
Then, using this transformation, find an infinite sequence of points on the
unit semicircle that are equally spaced in the sense of non-Euclidean
length.
5. Compute the cosine of the angle between the hyperbolic lines represented by
the semi
Homework 7 Solutions
June 1, 2015
6. (a) Assume for a contradiction that AB has two midpoints C and C . Then by denition
AC CB and AC BC . It follows that A C B and A C B. By assumption
=
=
C = C , and C and C are not equal to A or B. By B Axiom 3, either
Homework 3 Solutions
April 21, 2015
1. 2.1 If l and m are distinct lines that are not parallel, then l and m have a unique point in
common.
Proof.
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.
ix.
x.
xi.
xii.
xiii.
xiv.
xv.
l , m lines (l |m ) (P a point incident to
Homework 5 Solutions
May 13, 2015
22. (a) Suppose AB < CD and CD EF . By denition there is a point P between CD such
=
that AB CP . By Proposition 3.12 there exists a point Q between E and F such that
=
EQ CP . But then by Congruence Axiom 2, AB EQ which
Length / Angle measure Axioms:
Archimedes Axiom: If CD is a segment, A is any point and r is a ray with vertex A, then for
every point B A on r, there exists n N such that when CD is laid off n times along ray AB
(starting at A), then a point E is reached
Practice exam #1, Mat 141, Spring 2015.
Instructor: William Slofstra
Question 1 (14 points).
(a) Given a line L and a point P not on L, construct the perpendicular line to L through
P.
(b) Construct the bisecting ray through an angle with vertex ABC.
(c)
Practice nal exam, Mat 141, Spring 2015.
Instructor: William Slofstra
Question 1 (10 points). This question asks for ruler and compass constructions.
(a) Given three non-colinear points A, B, C in Euclidean space, construct a perpendicular
to AB through C
Homework 8 Solutions
June 8, 2015
Chapter 4
5. In this proof we use the fact that if X and Y are supplementary, then either X and Y
are congruent, and are right angles, or they are not congruent and one is obtuse and one is
acute. This follows from Theore
Practice nal exam: Answer key, Mat 141, Spring 2015.
Instructor: William Slofstra
Question 1: see homework from chapter 1.
Question 2.
(a) True, hyperbolic planes as evidenced by the Beltrami-Klein model.
(b) True, principle of duality.
(c) True, given a
MAT 141
Sample Midterm I
1) (a) (1 Point) State Euclids Postulate 4.
(b) (1 Point) State the negation of Euclids Postulate 4.
(c) (1 Point) Rewrite the following sentence using only the
logic words if, then, and, either, or, not, is, there is, are,
there
Practice exam #2, Mat 141, Spring 2015.
Instructor: William Slofstra
Question 1 (10 points). Consider the following two models of incidence geometry.
Points are lines in R3 through the origin. Lines are planes in R3 through the
origin. P is incident to H