LECTURE 21: MATCHING SETS (
§
9.39.4)
2 is not equal to 3, not even for large values of 2. Grabel’s Law
1.
OnetoOne
Recall that a relation from
A
to
B
is a function if each element of
A
occurs exactly once as a ﬁrst
coordinate in the relation. We are also interested what happens in the second coordinates.
One condition we could look for is whether second coordinates happen at most once (some of the
elements of
B
might not happen at all).
Deﬁnition 1.
A function
f
:
A
→
B
is
onetoone
or
injective
if any two elements of
A
map to distinct
elements of
B
. In other words, the elements of
B
occur at most once as second coordinates.
In other words:
∀
x,y
∈
A, f
(
x
) =
f
(
y
)
⇒
x
=
y
.
It is very easy to make functions that are not injective.
Example 2.
Let
A
=
{
1
,
2
}
and let
B
=
{
s,t
}
. Deﬁne
f
by the rule
f
(1) =
s
and
f
(2) =
s
. Then
f
is
not injective.
Example 3.
Is the function
f
:
R
→
R
deﬁned by
f
(
x
) = 7
x
+ 4 onetoone?
Proof.
Yes. Let
x,y
∈
R
and assume
f
(
x
) =
f
(
y
). Then 7
x
+ 4 = 7
y
+ 4 so 7
x
= 7
y
. Dividing by 7 we
get
x
=
y
.
±
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 Spring '11
 Smith
 Sets, Finite set, 4k, 5k, Injectivity

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