I am a beginner here and have recently started learning Discrete math. I dont know how to solve the following question.
Please help.
Q. Let A ={0,1,2,3} and B ={0,1,2}. How many functions are there from A to B that are onto?
I am a beginner here and have recently started learning Discrete math. I dont know how to solve the following question.
Please help.
Q. Let A ={0,1,2,3} and B ={0,1,2}. How many functions are there from A to B that are onto?
I will answer this question for the finite case only.
Being beginner, you may not understand the reply.
You need to know how to use the generalized inclusion/exclusion rule.
Let $\displaystyle |A|$ denote the number of elements in set $\displaystyle A$.
If $\displaystyle |A|<|B|$ there are no onto functions $\displaystyle A\to B$.
If $\displaystyle |A|=m\ge~n=|B|$ then number of onto functions $\displaystyle A\to B$ is given by:
$\displaystyle \text{Surj}(m,n)=\sum\limits_{j = 0}^n {\left( { - 1} \right)^j \dbinom{ n}{j} \left( {n - j} \right)^m } $.
Well, I did say that you may not have any idea what is going on.
It is rather advanced material. Here is webpage on inclusion/exclusion.
In essence we are selecting elements of $\displaystyle B$ to be included then excluded.