Solutions
MAT 300
(
H. Thieme
)
Test 3; April 21, 2006
1
2
3
4
5
∑
30
23
20
10
17
100
Work your problems in the space provided. Show all work clearly.
1.
Let
A
=
{
1
,
2
,
3
}
and consider the following relation
R
on
A
,
R
=
{
(1
,
2)
,
(2
,
1)
,
(3
,
2)
}
.
[30 points]
(a) Is
R
a function?
[3 points]
Yes, for every first component there is at most one second component.
(b) Dom(
R
)=
{
1
,
2
,
3
}
[3 points]
(c) Is
R
:
A
→
A
? Explain.
[3 points]
Yes, because Dom(
R
) =
A
.
(d) Ran(
R
)=
{
1
,
2
}
[3 points]
(e) Is
R
surjective (onto)? Explain.
[3 points]
No, because Ran(
R
)
6
=
A
.
(f) Is
R
injective (onetoone)? Explain.
[3 points]
No, because (1
,
2)
,
(3
,
2)
∈
R
.
(g)
R

1
=
{
(1
,
2)
,
(2
,
1)
,
(2
,
3)
}
.
[3 points]
(h) Is
R

1
a function? Explain.
[3 points]
No, because (2
,
1)
,
(2
,
3)
∈
R

1
.
(i)
R

1
(
{
1
,
2
}
) =
{
1
,
2
,
3
}
[3 points]
(j)
R
◦
R
=
{
(1
,
1)
,
(2
,
2)
,
(3
,
1)
}
[3 points]