A couple of example problems (Courtesy of D. Kaplan)
1. In the CPS85 dataset, we can explain some of the variation in wages by modeling it
with the variables sex and married. That is, there seems to be a relationship between
wages earned and sex and marit
Math 1180: Calculus II
Chapter 5 Practice Test
NAME
1. Write the parametric equations for a unit circle,
(a) Starting at (0,1) and going counterclockwise.
(b) Starting at (-1,0) and going clockwise.
vskip 0.8in
2. Using the first parametrization above, fi
Math 1170: Calculus I
Exam 3
NAME
Fully justify your answers. Among other things, this means write down all the steps you take
on the way to your answer.
1. (15) If f (1) = A, f (1) = V , g(1) = 1, and g (1) = H, find
(a) (f g) (1)
(b) (f g) (1)
+g
(c) (
Math 1170
EXAM 2 Practice Fall 2015
NAME
1. True or false. (Be ready to answer any of the TF problems on p 165 (Chapter 2 review)
(a)
(b)
(c)
If the function f (x) is continuous at x = 3 then limx3 f (x) exists.
If limx3 f (x)g(x) exists then the limit mu
Topics in Advanced Mathematics: Topology
Homework 7b
Wojciech Komornicki
3 April 2014
3.4 1 In the following, X is a subspace of R2 and x0 , x1 are points of X. Write down, if possible, explicit paths in
X joining x0 to x1 .
(i) X = (x, y) R2 | |x| > 1 or
Topics in Advanced Mathematics: Topology
Homework 5a
Solutions
4 March 2014
2.6 4 Prove that subspaces and (nite) products of spaces satisfying te rst axiom of countabillity also satisfy the
rst axiom of countability.
Proof: Let X satisfy the rst axiom of
Topics in Advanced Mathematics: Topology
Homework 8
Soluions
10 April 2014
3.4 8 Let X1 , X2 be subsets of X such that X = IntX1
component of X1 meets X2 .
IntX2 . Prove that, if X is path-connected, then each path
Proof: Since X is path-connected, it is
Topics in Advanced Mathematics: Topology
Homework 4b
Wojciech Komornicki
25 Febrauary 2014
2.5 1 Prove the continuity of the following functions : R3 R.
(i) (x, y, z) P (x, y, z) where P is a polynomial.
Proof:
We rst show that x P (x) where P is a polyno
Topics in Advanced Mathematics: Topology
Homework 10
Wojciech Komornicki
01 May 2014
4.4 1 Let x0 Sn and let be the hyperplane in Rn+1 which is perpendicular to the line determined by the origin
and x0 and which passes through the origin. Let s : Sn \ cfw
Topics in Advanced Mathematics: Topology
Homework 4a
Wojciech Komornicki
25 Febrauary 2014
2.4 1 Prove that the relation X is a subspace of Y is a partial order relation of topological spaces.
Proof:
Reexive Let X be a topological space. If A X is open th
Topics in Advanced Mathematics: Topology
Examination I
Solutions
11 March 2014
I. (36 points) Denitions Complete the following denitions:
a) Let X be a set and N X
a)
b)
c)
d)
(X). N is called a neighbourhood topology on X if
for every x X and N N (x), x
Topics in Advanced Mathematics: Topology
Homework 6a
Wojciech Komornicki
20 March 2014
2.8 2 Let f : R0 R0 be a continuous function such that
(i) f (x) = 0 x = 0,
(ii) x x f (x) f (x ),
(iii) f (x + x ) f (x) + f (x )
Let d be a metric on X. Show that the
Topics in Advanced Mathematics: Topology
Examination II
Solutions
17 April 2014
I. (24 points) Denitions Complete the following denitions:
a) Let X be a set. A metric d on X is a function d X X
R such that
i. d(x, y) 0 for all x, y X; d(x, y) = 0 if and
Topics in Advanced Mathematics: Topology
Homework 3b
Wojciech Komornicki
20 Febrauary 2014
2.3 1 State which of the following sets of points (x, y) of R2 are (a) open, (b) closed, (c) neither open nor closed.
(i)
(x, y) R2 : |x| < 1 and |y| < 1
Solution
T
Topics in Advanced Mathematics: Topology
Examination III
Solutions
13 May 2014
I. (24 points) Denitions Complete the following denitions:
a) Let X be a topological space. X is said to be compact if every open cover of X has a nite
subcover.
b) Let f X Y b
Topics in Advanced Mathematics: Topology
Homework 1 Solutions
Wojciech Komornicki
6 February 2014
1.1 1 Let a R and N be a subset of R. Prove that the following conditions are equivalent:
(a) N is a neighbourhood of a
(b) There is a > 0 such that [a , a +
Topics in Advanced Mathematics: Topology
Homework 3a Solutions
Wojciech Komornicki
18 Febrauary 2014
2.2 3 Let X be a topological space, and let A X. A point x in X is called limit point of A if each neighbourhood
of x contains points of A other than x. T
Topics in Advanced Mathematics: Topology
Homework 2b Solutions
Wojciech Komornicki
13 February 2014
2.1 1 Let X be a set with a neighbourhood topology. Prove that if A and B are subsets of X then,
IntA IntB = Int(A B)
IntA IntB Int(A B)
Proof: First, we s
Topics in Advanced Mathematics: Topology
Homework 2a Solutions
Wojciech Komornicki
11 February 2014
1.3 1 Prove that the Cantor set is uncountable.
Proof: The function dened in problem 1.3 2 is surjective. Since I is uncountable, K is uncountable.
1.3 2 P
Math 5890: Modern Algebra
HW 13
4 March 2013
p 152 #24 Let z0 U = cfw_z C | |z| = 1. Show that z0 U = cfw_z0 z | z U is a subgroup of U and compute
U/z0 U .
1
1
Proof: Note that 1 z0 U since 1 = z0 z0 and z0 U . If x, y z0 U there exist , U such
that x =
Math 5890: Modern Algebra
HW 11
1 March 2013
p 142 #1 The order of Z6 / 3 is 3.
p 142 #2 The order of (Z4 Z12 )/ 2 2 is 4.
p 142 #3 The order of (Z4 Z2 )/ (2, 1) is 4.
p 142 #4 The order of (Z3 Z5 )/cfw_0 Z5 is 3.
p 142 #5 The order of (Z2 Z4 )/ (1, 1) is
Math 5890: Modern Algebra
HW 10
27 February 2013
p 133 #16 If : S3 Z2 if given as in example 13.3, Ker() = cfw_(1), (1, 2, 3), (1, 3, 2).
p 133 #17 If : Z Z7 is dened by (1) = 4, then Ker() = cfw_7n | n Z, (25) = 2.
p 133 #18 If : Z Z10 is dened by (1) =
Math 5890: Modern Algebra
HW 9
22 February 2013
p 110 #6 The order of (3, 10, 9) in Z4 Z12 Z15 is 60.
p 110 #7 The order of (3, 6, 12, 16) in Z4 Z12 Z20 Z24 is 60.
p 110 #11 The subgroups of Z2 Z4 are cfw_0 cfw_0, cfw_0 cfw_0, 2, cfw_0 Z4 , Z2 cfw_0, Z2 c
Math 5890: Modern Algebra
HW 8
20 February 2013
p 102 #27 Let H be a subgroup of group G and let g G. Dene a one-to-one map of H onto Hg.
Proof: Dene : H Hg by (h) = hg and dene : Hg H by (a) = ag 1 .
Since and are inverses of each other, is one-to-one an
Math 5890: Modern Algebra
HW 7
18 February 2013
p 94 #1 The orbits are cfw_1, 5, 2, cfw_3, cfw_4, 6
p 94 #2 The orbits are cfw_1, 5, 8, 7, cfw_2, 6, 3, cfw_4
p 94 #3 The orbits are cfw_1, 2, 3, 5, 4, cfw_6, cfw_7, 8
p 94 #10 (1, 8)(3, 6, 4)(5, 7)
p 94 #11
Math 5890: Modern Algebra
HW 5
11 February 2013
p 58 #43 Show that if H and K are subgroups of an abelian group G, then
cfw_hk | h H and k K
is a subgroup of G.
Proof: Let G be an abelian group and H and K subgroups of G.
Let HK = cfw_hk | h H and k K. No
Math 5890: Modern Algebra
Solutions
Examination II
3 April 2013
1. (10 points) Complete the following denition:
Let G be a group and p be a prime. A subgroup H of G is called a p-Sylow subgroup of G if it is a
maximal p-subgroup of G.
Note: A subgroup is
Math 5890: Modern Algebra
HW 4
8 February 2013
p 48 #31 If is a binary operation on a set S, an element x of S is an idempotent for if x x = x.
Prove that a group has exactly one idempotent element.
Proof: Clearly the identity element e is an idempotent.