1. ## Surjective, Injective, Bijective

Hello,

I would like to pose a question concerning the definitions for the functions mentioned in the title.

In my book I found the following definitions:

Function- A relation R from A to B is a function iff:

1) Each element in the domain is paired with just one element in the range.

2) The domain of R is equal to A.

Surjective onto function-

If every element in the range is paired with at least one element in the domain.

Injective one to one function-

If every element in the range is paired with exactly one element in the domain.

Bijective function-

Both Surjective and Injective.

Now here is the question:

Is the example for an injective function that is not Surjective and therefore not Bijective achieved when there are more elements in the domain than in the range? Are there any other possibilities for having an injective function?

Than for example we have A= { a, b, c} B = { 1, 2}

And a and b are paired with 1 whereas c with 2.

That's a function because every element of A is paired with one element of B. That surjective because every element of B is paired with at least one element of A.

2. Originally Posted by feliks0
Now here is the question:

Is the example for an injective function that is not Surjective and therefore not Bijective achieved when there are more elements in the domain than in the range? Are there any other possibilities for having an injective function?
Than for example we have A= { a, b, c} B = { 1, 2}
And a and b are paired with 1 whereas c with 2.
That's a function because every element of A is paired with one element of B. That surjective because every element of B is paired with at least one element of A.
What is the question?
Is it about injections from A to B? There are none.

3. If we have the following ordered pairs:

{<a,1>, <b,2>}

Is that an Injective function which is not surjective?

If not, I would like an example of such a function please.

4. Originally Posted by feliks0
If we have the following ordered pairs:
{<a,1>, <b,2>}
Is that an Injective function which is not surjective?
If not, I would like an example of such a function please.
It is not even a function.
Any function from A to B must have three pairs because A has three elements.
Because B has only two terms, there are no injections from A to B.
Likewise there are no surjections from B to A.

5. So could you please give me an example of an injective function which is not also surjective?

6. Originally Posted by feliks0
So could you please give me an example of an injective function which is not also surjective?
We must change the sets.
$C=\{a,b,c\}~\&~D=\{1,2,3,4\}$
$\phi :C \mapsto D,\quad \phi = \left\{ {(a,4),(b,2),(c,3)} \right\}$
That is an injection which is not a surjection.