Section 2.4
(b) [BB]
I
=
{a
I
a'"
I}
=
{a
I
I
E
Q}
=
{a
I
a
E
Q}
=
Q"
{O}.
(c) [BB]
1/
=
~
=
2 E Q, so y'3 '"
v'l2
and hence y'3
=
v'l2.
41
6. Reflexive: For any
a
E
N,
a'" a
since
a
2
+a
=
a(a+
1) is even, as the product of consecutive natural
numbers.
Symmetric: If
a
'"
b,
then
a
2
+
b
is even.
It
follows that either
a
and
b
are both even or both are odd.
If they are both even,
b
2
+
a
is the sum of even numbers, hence, even.
If
they are both odd,
b
2
+
a
is
the sum of odd numbers and, hence, again, even. In both cases
b
2
+
a
is even, so
b
'"
a.
Transitive: If
a
'" band
b
'" c, then
a
2
+
b
and
b
2
+ c are even, so
(a
2
+
b)
+
(b
2
+ c) is even; in
other words,
(a
2
+ c) +
(b
2
+
b)
is even. Since
b
2
+
b
is even,
a
2
+ c is even too; therefore,
a
'" c.
The quotient set is the set of equivalence classes. Now
2.
{evens
a=
{x
I
x
+alseven}
=
odds
So
A/",=
{2Z, 2Z
+
I}.
if
a
is even
if
a
is odd
7. (a) [BB] Reflexive: For any
a
E R,
a
'"
a
because
a

a
=
0 E Z.
Symmetric:
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 Summer '10
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 Graph Theory, c., 5 m, 4k, 4 k

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