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46
Solutions to Exercises
21. (a) [BB] The ordered pairs defined by'" are (1,1), (1,4), (1,9), (2,2), (2,8), (3,3), (4,1), (4,4),
(4,9), (5,5), (6,6), (7,7), (8,2), (8,8), (9,1), (9,4), (9,9).
(b) [BB]
I = {1,4,9} = 4 = 9;2 = {2,8} = 8;
'3
= {3};"5 = {5}; 6 = {6};
'7
= {7}.
(c) [BB] Since the sets {1, 4, 9}, {2, 8}, {3}, {5}, {6} and {7} partition
A,
they determine an equiv
alence relation, namely, that equivalence relation in which
a
'"
b
if and only if
a
and
b
belong to
the same one of these sets. This is the given relation.
22. [BB] Reflexive:
If
a
E
A,
then
a
2
is a perfect square, so
a
'"
a.
Symmetric:
If
a
'"
b,
then
ab
is a perfect square. Since
ba
=
ab, ba
is also a perfect square, so
b
'"
a.
Transitive:
If
a
'"
b
and
b
'" e, then
ab
and
be
are each perfect squares. Thus
ab
=
x
2
and
be
=
y2
x
2
y2
(xy)2
for integers
x
and
y.
Now
ab
2
e
=
x
2
y2,
so
ae
=
~
=
b
.
Because
ae
is an integer, so also
x:
is an integer. Therefore,
a
'"
e.
23. (a) The order pairs
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 Summer '10
 any
 Graph Theory

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