MAT 300 Spring 06
Dr. Zandieh
Homework #8
Solutions
1. Consider the family of sets consisting of all parabolas of the form where a
is a real number.
axy += 2
a. Prove that this set of parabolas forms a partition of the plane,
ℜ
2.
Let and let
ℜ∈a
()
{
}
axyyxAa+=ℜ∈= 22 |, . Define the family . We
wish to prove that F is a partition of
{}
ℜ∈
=a
a
AF
ℜ
2.
(i) First, note that (0, a) since
2
ℜ∈
ℜ
∈
0 and
ℜ
∈
a. Also, , so (0,
a) . Thus, aa += 2
0
a
A∈
φ
≠
∈∀ aa AFA , . So, F satisfies the first property of a
partition.
(ii) Next, suppose FAA ba
∈
, and
φ
≠
∩
ba AA . Then there exists
by the negation of the definition of the empty set. Thus,
and
ba AAyx ∩∈),(
a
Ayx ∈),(b
Ayx
∈
),( by definition of intersection. By definition of ,
, or . Similarly, or by definition
of . So, by the symmetric and transitive properties of equality,
, so
a
A
axy += 2axy =− 2bxy += 2bxy =− 2
b
A
bxya =−= 2ba
=
. Thus, ba AA
=
, and F satisfies the second property
of a partition.
(iii) Finally, we wish to show that . So, first let . Then
or some
2
ℜ=
ℜ∈
U
aa
AU
ℜ∈
∈
aa
Ayx ),(
a
Ayx ∈),( f
ℜ
∈
a by definition of the union of a family of sets.
Thus, by definition of , . Now, let . Consider the
set : note that , so
a
A2
),( ℜ∈yx 2
),( ℜ∈yx
2
xy
A−)(22 xyxy −+= 2
),( xy
Ayx −
∈
. Since is closed
under subtraction and multiplication, , so . Thus,
by the definition of the union of a family of sets. So,
if and only if , and we have that by
definition of set equality. Thus, F satisfies the third property of a partition.
ℜ
ℜ∈− 2
xy FA xy ∈
−2
U
ℜ∈
∈
aa
Ayx ),(
U
ℜ∈
∈
aa
Ayx ),(2
),( ℜ∈yx 2
ℜ=
ℜ∈
U
aa
A
Since F satisfies all three properties, F is a partition of
ℜ
2.
b) Write an equivalence relation that will partition the plane into parabolas of the form
where a is a real number.
axy += 2
We want two points and in
),( yx ),( vu
ℜ
2 to be equivalent if and only if they are on the
same parabola, i.e. and . In other words, we want and
axy += 2auv += 2axy =− 2
