# Thread: Number of everywhere defined and one to one functions? Help please?

1. ## Number of everywhere defined and one to one functions? Help please?

Let A have 5 elements and let B have 4 elements.

1) How many everywhere defined functions are there from A to B?
For the function to be everywhere defined it must contain all 5 elements of A and each can have 4 possible partners so there are 4*4*4*4*4=4^5 possible everywhere defined functions?Is this correct?If not how do I do it?

2) How many one to one functions are there from A to B?
Logically the function can have a maximum of 4 elements but I don't know how to count the possibilities here.

Any help would be appreciated.

2. Originally Posted by chocaholic
Let A have 5 elements and let B have 4 elements.

1) How many everywhere defined functions are there from A to B?
For the function to be everywhere defined it must contain all 5 elements of A and each can have 4 possible partners so there are 4*4*4*4*4=4^5 possible everywhere defined functions?Is this correct?If not how do I do it?

It is correct

2) How many one to one functions are there from A to B?
Logically the function can have a maximum of 4 elements but I don't know how to count the possibilities here.

[color=red]There is none since at least one element of A must be mapped to some element of B which is already the image of some other element, because there are 5 elements in A but only 4 in .

Tonio

Any help would be appreciated.

.

3. Thanks with 1 but with 2 the function does not have to be everywhere defined just one to one so it can consist of 1 or 2 or 3 or 4 elements of A and not necessarily all 5 as long as it is related to only one element of B and the elements of B are not repeated?

4. Originally Posted by chocaholic
Thanks with 1 but with 2 the function does not have to be everywhere defined just one to one so it can consist of 1 or 2 or 3 or 4 elements of A and not necessarily all 5 as long as it is related to only one element of B and the elements of B are not repeated?

When you say "functions from A to B" it is usually understood that the functions are defined in the whole A, otherwise there's no point in mentioning A , is it? That's what I meant.

Tonio