Math 150a: Modern Algebra
Homework 5
This problem set is due Wednesday, October 31. Do problems 2.4.19, 2.5.3 (equivalently, the number of
partitions), and 2.6.10(a), in addition to the following:
GK1.
Set arithmetic can be interesting even when the sets involved are not subgroups or cosets. The first
two parts of this problem involve the vector space
R
2
, which among other things is an additive group.
a.
Let
T
=
{
x
,
y
,
1
−
x
−
y
≥
0
} ⊂
R
2
be the filledin triangle with vertices at
(
0
,
0
)
,
(
1
,
0
)
, and
(
0
,
1
)
.
Find
T
−
T
in set arithmetic.
b.
Let
C
=
{
x
2
+
y
2
=
1
} ⊂
R
2
be the unit circle. Find
C
+
3
C
in set arithmetic. (Here 2
C
does not
mean
C
+
C
+
C
, but rather 3 times each element of
C
.)
c.
If
G
is a group, then multiplication of subsets of
G
is itself associative. (You are not required to
prove that.) Does the semigroup of subsets of
G
have an identity? Which subsets have inverses?
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This homework help was uploaded on 02/01/2008 for the course MATH 150A taught by Professor Kuperberg during the Spring '03 term at UC Davis.
 Spring '03
 Kuperberg
 Algebra, Addition

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