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SOLUTIONS FOR QUIZ #1.
1(a) Prove that the set
S
=
{
(
x,y
) : 1
< x
2
+
y
2
}
in
R
2
is open. You
have to give a rigorous proof based on the de nition of an
open set.
We have to show that
∀
x
∈
S,
∃
r >
0
such that
B
(
r,
x
)
⊂
S.
Let
r
=

x
 
1
.
Then
∀
y
∈
B
(
r,
x
) =
{
x

y

< r
}
we have

y
 ≥ 
x
  
x

y

>

x
 
r
=

x
  
x

+ 1 = 1
(by exercise #6, sec.1.1, a corollary to triangle inequality

a
  
b
 ≤ 
a

b

)
so

y

>
1
and
y
∈
S
meaning that
B
(
r,
x
)
⊂
S.
(b) Give an example of a set
S
such that the interior of
S
is unequal
to the interior of the closure of
S.
Justify your answer.
Let
S
=
{
(
x,y
) : 0
< x
2
+
y
2
<
1
} ⇒
S
=
S
int
¯
S
=
{
(
x,y
) :
x
2
+
y
2
≤
1
} ⇒
(
¯
S
)
int
=
{
(
x,y
) :
x
2
+
y
2
<
1
} 6
=
S
int
(other examples possible)
2(a) Show directly from the de nition that the set
S
=
{
(
x,y
)
∈
R
2
:
xy
= 1
}
is disconnected. That is produce a disconnection (in
a set notation) for
S.
S
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 Fall '09
 RomauldStanczak
 Calculus

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