Math 113 Homework 1 – Solutions
1. (a) Let
f
:
X
→
Y
and
g
:
Y
→
Z
. Show that if
g
◦
f
:
X
→
Z
is onto, then
g
is onto.
Assume
g
◦
f
is onto, and let
z
be an element of
Z
. Since
g
◦
f
is onto, there exists
x
∈
X
such that
g
◦
f
(
x
) =
z
. Let
y
=
f
(
x
)
∈
Y
: then
g
(
y
) =
g
(
f
(
x
)) =
z
. Therefore, for every
z
∈
Z
there exists
y
∈
Y
such that
g
(
y
) =
z
. Hence
g
is onto.
Show that if
g
◦
f
is onetoone, then
f
is onetoone.
Assume
g
◦
f
is onetoone, and let
x
1
, x
2
∈
X
be such that
f
(
x
1
) =
f
(
x
2
).
Then
g
◦
f
(
x
1
) =
g
(
f
(
x
1
)) =
g
(
f
(
x
2
)) =
g
◦
f
(
x
2
), and since
g
◦
f
is onetoone we deduce that
x
1
=
x
2
. Hence
f
is onetoone.
(b) Show that
f
:
X
→
Y
is bijective (i.e., onetoone and onto) if and only if
there exists
g
:
Y
→
X
with
g
◦
f
= id
X
and
f
◦
g
= id
Y
.
First we show that, if
f
is bijective, then there exists
g
with the stated properties. Given
a bijective map
f
:
X
→
Y
, define
g
:
Y
→
X
as follows. Given
y
∈
Y
, there exists a unique
x
∈
X
such that
f
(
x
) =
y
(since
f
is bijective), and we define
g
(
y
) =
x
. Given
x
∈
X
, let
y
=
f
(
x
), then by definition
g
(
y
) =
x
so
g
(
f
(
x
)) =
x
. Hence
g
◦
f
= id
X
. Given
y
∈
Y
, let
x
=
g
(
y
), by definition
f
(
x
) =
y
, so
f
(
g
(
y
)) =
y
. Therefore
f
◦
g
= id
Y
.
Conversely, assume there exists
g
:
Y
→
X
such that
f
◦
g
= id
Y
and
g
◦
f
= id
X
. Then
g
◦
f
= id
X
is onetoone, so by part (a)
f
is onetoone. Moreover
f
◦
g
= id
Y
is onto, so by
part (a)
f
is onto. Hence
f
is a bijection.
(Optional: what happens if you drop one of the two conditions on
g
?)
If we can find
g
:
Y
→
X
such that
g
◦
f
= id
X
, then by part (a)
f
is onetoone.
However
f
is not necessarily onto: for example consider
X
=
{
1
}
,
Y
=
{
1
,
2
}
,
f
(1) = 1,
g
(1) =
g
(2) = 1. Similarly, if we can find
g
such that
f
◦
g
= id
Y
then by (a)
f
is onto, but
it is not necessarily onetoone (same counterexample, exchanging
X
with
Y
and
f
with
g
).
2. Fraleigh, section 0, exercises 29–34. Determine whether the given relations
are equivalence relations; describe the partition arising from each equivalence
relation.
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 Spring '08
 OGUS
 Math, Equivalence relation, equivalence classes, binary expansion

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