Solutions to Week 2 Homework problems from Abbott
Problems (section 1.2) 1.3.1, 1.3.4, 1.3.6, 1.3.9, 1.4.2, 1.4.5, 1.4.11, 1.5.8
1.3.1
Note: This exercise is good practice for abstract algebra too!
(a) Show that, given any element
z
∈
Z
5
, there exists an element
y
such
that
z
+
y
= 0. The element
y
is called the
additive inverse
of
z
.
(b) Show that, given any
z
= 0 in
Z
5
, there exists an element
x
such that
zx
= 1. The element
x
is called the
multiplicative inverse
of
z
.
(c) Investigate the set
Z
4
=
{
0
,
1
,
2
,
3
}
(where addition and multiplication
are defined modulo 4) for the existence of additive and multiplicative inverses.
Make a conjectuve about the values of
n
for which additive inverses exist in
Z
n
, and then form another conjecture about the existence of multiplicative
inverses.
(a) 0 + 0 = 0; 1 + 4 = 0; 2 + 3 = 0 (addition is mod 5). Hence we have
found additive inverses for each element of
Z
5
.
(b) 1
×
1 = 1; 2
×
3 = 1; 4
×
4 = 1 (multiplication mod 5). Hence we
have multipliative inverses for each nonzero element.
(c) 0 + 0 = 0; 1 + 3 = 0; 2 + 2 = 0 (all mod 4) hence additive inverses
exist. But, 1
×
1 = 1; 3
×
3 = 1 (mod 4) but 2 does not have a multiplicative
inverse. In fact,
n
needs to be a prime for
Z
n
to be a field.
1.3.4
Assume that A and B are nonempty, bounded above, and satisfy
B
⊆
A
. Show that sup
B
≤
sup
A
.
Proof: By definition of sup, we have sup
A
≥
a
for all
a
∈
A
, and sup
A
is
the smallest number with this property. Suppose sup
B >
sup
A
. Then sup
A
cannot be an upper bound of
B
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 Spring '06
 Spielberg
 Algebra, Rational number

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