Wednesday, 3 April 2019

10:20 AM

Suppose that  is a set with  elements and  is a set with  elements.

 

1. How many functions  are there?

 

First we count all possible functions from  to

Choose   …  ways

Choose   …  ways

…

Choose   …  ways

 

There are  ways

 

2. How many of these are one-to-one?

No repetitions are allowed…

This is equal to zero, as there are less output elements than input elements

 

3. How many are onto?

Write

 

Let  be the set of all functions from  to  where  is missing

for

 

 

 

 

 

 

 

4. How many are bijections?

If  then , else