One-to-One/Onto Functions
Here are the definitions:

is one-to-one (injective) if maps every element of to a unique element in .
In other words no element of are mapped to by two or more elements of .
.


is onto (surjective)if every element of
other words, nothing is left out.
. In
.


is mapped to by some element of
is one-to-one onto (bijective) if it is both one-to-one and onto. In this case the
map is also called a one-to-one correspondence.
Example-1
Classify the following functions
onto.
between natural numbers as one-to-one and
One-to-One?
Onto?
Yes
No
Yes
No
No
Yes
Yes
Yes
.
It helps to visualize the mapping for each function to understand the answers.
Reasons

is not onto because it does not have any element
instance.

is not onto because no element

is not one-to-one since
such that
such that
, for
, for instance.
.
Example-2
Prove that the function
is one-to-one.
Proof: We wish to prove that whenever
then
for two numbers
. Therefore,
. Splitting cases on , we have
 For
,
, therefore
for this case.
. Let us assume that
. Which means that

For
, we have
. Therefore, it follows that
for both cases.
Example-3
Prove that the function
is onto.
Proof
Given any
, we observe that
are mapped onto.
is such that
. Therefore, all
Claim-1 The composition of any two one-to-one functions is itself one-to-one.
Proof
Let
and
be both one-to-one. We wish to tshow that
is also
one-to-one.
 Assume that
for two elements
.

Therefore
.

Since is itself one-to-one, it follows that

Since is one to one and

Therefore
.
it follows that
can happen only if
The reasoning above shows that
.
.
is one-to-one.
Claim-2 The composition of any two onto functions is itself onto.
Proof
Let
and
be onto functions. We will prove that
 Let
be any element.

Since is onto, we know that there exists

Likewise, since is onto, there exists

Combining,

Thus,
is onto.
.
is also onto.
such that
such that
.
.