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Section 2.4
45
Transitive:
If
(a, b)
f'V
(c, d)
and
(c, d)
f'V
(e, I),
then
a
+
2b
=
c
+
2d
and c
+
2d
=
e
+
2/, so
a
+
2b
=
e
+
2/ and
(a, b)
f'V
(e,l).
The quotient set is the set of equivalence classes. We have
(a, b)
=
{(x, y)
I
(x, y)
f'V
(a, b)}
=
{(x, y)
I
x
+
2y
=
a
+
2b}
=
{(x, y)
I
y

b
=
!(x

a)}
which describes the line through
(a, b)
with slope
!.
The quotient set is the set of lines with
slope
!.
(b) This is an equivalence relation.
Reflexive:
If
(a, b)
E R2, then
ab
=
ab,
so
(a, b)
f'V
(a, b).
Symmetric:
If
(a, b)
f'V
(c,
d),
then
ab
=
cd
so
cd
=
ab
and
(c,
d)
f'V
(a, b).
Transitive:
If
(a, b)
f'V
(c, d)
and
(c, d)
f'V
(e, I),
then
ab
=
cd
and
cd
=
e/,
so
ab
=
cd
=
e/,
ab
=
e/
and
(a, b)
f'V
(e,
f).
The quotient set is the set of equivalence classes. We have
(a, b)
=
{(x,y)
I
(x,y)
f'V
(a,b)}
=
{(x,y)
I
xy
=
ab}
and consider two cases.
If
either
a
= 0 or
b
= 0, then
(a, b)
=
{(x, y)
I
xy
= O}; that is,
{(x, y)
I
x
=
0 or
y
=
O}. Hence,
(a,
b)
is the union of the xaxis and the yaxis. On the other
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 Summer '10
 any
 Graph Theory

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