Section
4.1
2.
(a) [BB] True.
If
a
and
b
are real numbers, certainly
a

b
is a real number.
(b) False.
If
a
=
1,
b
=
2,
then
a

b
=
1
but
b

a
=
1.
75
(c) False.
Ifa
=
1,b
=
2,c
=
3, then
(ab)
c
=
13
=
4,
while
a
(bc)
=
1
(1)
=
2.
3. Let
a
=
3,
b
=
4, c
=
2.
Then 3
:5
4 but 3(
2)
=
6
1:.
8
=
4(
2).
4.
(a) [BB]
q
=
29,
r
=
7;
(b)
[BB]
q
=
30,
r
=
10;
(c) [BB]
q
=
29,
r
=
7;
(d)
[BB]
q
=
30,
r
=
10.
5.
(a)
q
=
278,
r
=
4;
(d)
q
=
279, r
=
15;
6.
(a)
q
=
316,
r
=
21;
(d)
q
=
20, r
=
2984;
(b)
q
=
279,
r
=
15;
(e)
q
=
0,
r
=
19;
(b)
q
=
29, r
=
972;
(e)
q
=
55, r
=
134;
(c)
q
=
278,
r
=
4;
(0
q
=
1,
r
=
5267.
(c)
q
=
70, r
=
613;
(0
555,555,123
=
555,555,555 
432
=
5(111,111,111) 
432
=
4(111,111,111) + 111,111,111 
432
=
4(111,111,111) + 111,110,679
So
q
=
4; r
=
111,110,679
(g)
q
=
2,104,000; r
=
0
7. [BB] Write
a
=
3q
+ r with r
=
0,1,2.
Case 1:
r
=
O.
In this case,
a
=
3q,
so
a
2
=
9q2
=
3k
with
k
=
3q2.
Case 2:
r
=
1.
Here,
a
=
3q
+ 1, so
a
2
=
9q2
+
6q
+ 1
=
3k
+ 1 with
k
=
3q2
+
2q.
Case
3:
r
=
2.
In this case,
a
=
3q
+ 2 and
a
2
=
9q2
+
12q
+ 4
=
3k
+ 1 with
k
=
3q2
+
4q
+
1.
S.
(a) [BB] The domain
of
f
is
Z;
its range is Z as well, because given
q
E
Z,
we have
q
=
f(qn).
(b)
[BB]
f
is not onetoone since
f(O)
=
f(l).
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 Summer '10
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 Real Numbers, Graph Theory

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