Math 301, Homework 1
Solutions
1. Prove that for a function
f
:
S
!
T
is onetoone if and only if for all
subsets
C
1
; C
2
in
S
we have
f
(
C
1
\
C
2
) =
f
(
C
1
)
\
f
(
C
2
)
:
Solution 1
Suppose
f
is 12. Take two sets
C
1
; C
2
in
S:
Let
y
2
f
(
C
1
\
C
2
)
:
Then there is some
x
2
C
1
\
C
2
satisfying
f
(
x
) =
y:
But then
x
2
C
1
and
x
2
C
2
;
so
y
=
f
(
x
)
2
f
(
C
1
)
and
y
=
f
(
x
)
2
f
(
C
2
) ;
i.e.
y
2
f
(
C
1
)
\
f
(
C
2
)
::
This shows that
f
(
C
1
\
C
2
)
±
f
(
C
1
)
\
f
(
C
2
)
:
Conversely let
y
2
f
(
C
1
)
\
f
(
C
2
)
:
Then there are some
x
j
2
C
j
;
j
= 1
;
2
satisfying
f
(
x
j
) =
y:
Since
f
is 11, we must have
x
1
=
x
2
:
Then
x
1
=
x
2
2
C
1
\
C
2
;
so
y
=
f
(
x
j
)
2
f
(
C
1
\
C
2
)
:
This shows
that
f
(
C
1
)
\
f
(
C
2
)
±
f
(
C
1
\
C
2
)
:
Then
f
(
C
1
\
C
2
) =
f
(
C
1
)
\
f
(
C
2
)
:
Conversely suppose for all subsets
C
1
; C
2
in
S
we have
f
(
C
1
\
C
2
) =
f
(
C
1
)
\
f
(
C
2
)
:
Suppose
f
(
x
1
) =
f
(
x
2
) =
y
for some
x
1
6
=
x
2
:
Let
C
j
=
f
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 Fall '08
 ALBERTERKIP
 Math, Sets, Inverse function, Injective function, Basic concepts in set theory, Identity element

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