MATH 5320
Exam I
October 4, 2004
In-Class Make-Up
Due Friday, Oct 8
Answer the problems on separate paper. You do not need to rewrite the problem statements on your
answer sheets. Work carefully. Do your own work. Show all relevant supporting steps!
1. (10) Let z = −2 3 i − 2 and w = 4 − 4i . Write in rectangular form, a + b i, and polar form,
r cis θ , each of the following:
3
a.
z2
w3
b.
 z+ w


 z+ w
2. (10) Prove Proposition 1.13 d.
3. (22) Give examples of sets in £ with the ususal topology:
a.
b.
i)
A set A such that int ( A \int A ) = ∅
ii)
A set A such that int ( A \int A ) ≠ ∅
i)
ii)
iii)
A set A such that A has only finitely many components
A set A such that A has countably infinitely many components
A set A such that A has uncountably many components
A countable collection{ An }∞n =1 is said to be distinct if j ≠ k ⇒ A j ≠ Ak for all j , k ∈ ¥
c.
i)
A countable collection of distinct closed sets {Gn }∞n =1 such that
∞
a)
U Gn is open ( ≠ £ )
n =1
∞
b)
U Gn is closed ( ≠ £ )
n =1
∞
c.
U Gn is neither open nor closed
n =1
ii)
A countable collection of distinct open sets {Fn }∞n =1 such that
∞
a)
I Fn is closed ( ≠ ∅)
n =1
∞
b)
c.
I Fn is open (≠ ∅ )
n =1
∞
I Fn is neither open nor closed
n =1
4. (10) Prove the following proposition: Let ( X , d ) be a metric space. Let f , g : X → £ be
uniformly continuous on X. Then, f + g is uniformly continuous on X.
5. (9) Give examples of sequences { xn } , { y n } in £ such that
a.
lim xn y n = L , lim x n = M , lim yn does not exist
b.
lim xn yn = L , lim xn does not exist, lim yn does not exist
c.
lim
xn
M
= L , lim xn = M , lim y n = N , L ≠
yn
N
6. (6) Give examples of distinct sequences { xn } in £ such that
a.
# { xn }' = 5 (i.e., the sequence has exactly five limit points}
b.
the sequence { xn } is given by a formula xn = f ( n ) , where f : ¥ → £ , such that
x1 = 1, x2 = 2, x3 = 4, x4 = 8, x5 = 16, but x 6 ≠ 32
(i.e., give a closed formula for the function f defining the sequence xn ).
7. (8) For a, b, c ∈ £ , a ≠ 0 , show that ab = ac ⇒ b = c .
8. (6) Provide a counterexample to each of the following assertions:
a.
In a metric space ( X , d ) , if a set A is closed and bounded, then A is compact.
b.
Let ( X , d ) , ( Ω, ρ ) be metric spaces. Let f : X → Ω . If f is uniformly continuous
on X, then f is Lipschitz on X.
9. (20) Classification Problem. Correctly identify whether the following subsets of £ are: (a) open; (b)
closed; (c) connected; (d) polygonally path connected; (e) compact; (f) complete; (g) bounded;
(h) region. You do not need to provide a rationale for your classification. Fill out the
classification information on the attach table.
A.
x 2 y2
B (0,5)\ E where E = {z = x + iy :
+
< 1}
25 4
B.
B (0,5)\ E where E = {z = x + iy :
C.
B (0,1)\ B ( 1 2 , 1 2 )
D.
{B (0,4)\ B (1,1)} ∩ {z : R e z ≥ 0}
∞
E.
Ul
n =1
n
, where each l = [0,
n
x 2 y2
+
< 1}
25 4
π
)
2n ]
n
cis(
Classification Table for Problem 9
open
A
B
C
D
E
closed
connected
polygonally
path
connected
compact
complete
bounded
region