# Thread: Partitions for surjective functions

1. ## Partitions for surjective functions

Let A be the set of all functions f:{1,2,3,4,5}->{1,2,3} and for i=1,2,3 let Ai denote a subset of the functions f:{1,2,3,4,5}->{1,2,3}\i.

i)What is the size of :
1). A,
2).the sizes of its subsets Ai,and
3).AiAj (i<j) also
4).A1A2A3.

ii)Find with justification the number of surjective functions f:{1,2,3,4,5}->{1,2,3}.

iii)Explain how each surjective function gives a 3-partiton of f:{1,2,3,4,5}

iv)How many 3-partitons of f:{1,2,3,4,5} are there altogethere?

i)1).3^5........2).2^5 each 3).1^5 each.......4).0
ii)We can use stirling numbers of the second kind to find this out S(5,3).
iii)Because the image of the function is made up of 3 elements?
iv)I don't know...I"m still thinking the same as for part ii)

2. ## Re: Partitions for surjective functions

Some notation. If $f:A\to B$ and $C\subseteq B$ then $f^{-1}[C]=\{x\in A:~f(x)\in C\}$.

It can be shown that if $f$ is a surjection then the collection $\left\{ {f^{ - 1} \left[ {\{ y\} } \right]:{\kern 1pt} y \in B} \right\}$ is a partition of $A$

3. ## Re: Partitions for surjective functions

ah...yes...there 3 in total .....I have the partitions properties in my head I can't describe them here because of the LaTex.

4. ## Re: Partitions for surjective functions

Originally Posted by StefanM
ah...yes...there 3 in total .....I have the partitions properties in my head I can't describe them here because of the LaTex.
Well there are more than three.
There are 150 surjections from a set of five to a set of three.
Each of those surjections defines a three-cell partition of the set of five.

If $\phi:\{1,2,3,4,5\}\to\{1,2,3\}$ is a surjection then $\left\{ {\phi ^{ - 1} (\{ 1\} ),\phi ^{ - 1} (\{ 2\} ),\phi ^{ - 1} (\{ 3\} )} \right\}$ is a partition of $\{1,2,3,4,5\}$