40
2.5
Let
A
and
B
be sets, and let
a
∈
A
and
b
∈
B
. De
fi
ne their
ordered pair
as follows:
(
a
,
b
)
= {
a
,
{
a
,
b
}}
.
If
a
∈
A
and
b
∈
B
, prove that
(
a
,
b
)
=
(
a
,
b
)
if and only if
a
=
a
and
b
=
b
.
Solution.
The result is obviously true if
a
=
a
and
b
=
b
.
For the converse, assume that
{
a
{
a
,
b
}} = {
a
{
a
,
b
}}
There are two cases:
a
=
a
and
{
a
,
b
} = {
a
,
b
};
a
= {
a
,
b
}
and
{
a
,
b
} =
a
.
If
a
=
a
, we have
{
a
,
b
} = {
a
,
b
} = {
a
,
b
}
. Therefore,
{
a
,
b
} − {
a
} = {
a
,
b
} − {
a
}
.
If
a
=
b
, the left side is empty, hence the right side is also empty, and so
a
=
b
; therefore,
b
=
b
. If
a
=
b
, the the left side is
{
b
}
, and so the right
side is nonempty and is equal to
{
b
}
. Therefore,
b
=
b
, as desired.
In the second case,
a
= {
a
,
b
} = {{
a
,
b
}
b
}
. Hence,
a
∈ {
a
,
b
}
and
{
a
,
b
} ∈ {{
a
,
b
}
,
b
} =
a
,
contradicting the axiom
a
∈
x
∈
a
being false. Therefore, this case cannot
occur.
2.6
Let
= {
(
x
,
x
)
:
x
∈
R
}
; thus,
is the line in the plane which passes
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 Fall '11
 KeithCornell
 perpendicular bisector, p p

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