# Thread: function composition and inverse function

1. ## function composition and inverse function

Let A={1,2,3,4,5} and B={6,7,8,9,10,11,12}.
How many functions f:A->B are such that f -1({6,7,8})={1,2}?

2. Originally Posted by sbankica
Let A={1,2,3,4,5} and B={6,7,8,9,10,11,12}.
How many functions f:A->B are such that f -1({6,7,8})={1,2}?
Any function $\phi :\{ 1,2\} \mapsto \{ 6,7,8\}$ is such that $\phi ^{ - 1} \{ 6,7,8\} = \{ 1,2\}$.
There are $3^2$ such functions. WHY?

How many functions $\{ 3,4,5\} \mapsto \{ 9,10,11,12\}?$

Can you answer the question posted?

3. 4^3?

5. Originally Posted by sbankica
You got it!

6. Originally Posted by Plato
Any function $\phi :\{ 1,2\} \mapsto \{ 6,7,8\}$ is such that $\phi ^{ - 1} \{ 6,7,8\} = \{ 1,2\}$.
There are $3^2$ such functions. WHY?

How many functions $\{ 3,4,5\} \mapsto \{ 9,10,11,12\}?$

Can you answer the question posted?
PLATO, you have been so helpful.

I see the following six functions:
f(1)=6; f(1)=7; f(1)= 8; f(2)=6; f(2)=7; f(2)=8

What are the other three functions?

Thank you.

7. Originally Posted by oldguynewstudent
I see the following six functions:
f(1)=6; f(1)=7; f(1)= 8; f(2)=6; f(2)=7; f(2)=8
What are the other three functions?
$\left\{ {\left( {1,6} \right),\left( {2,6} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,8} \right)} \right\}$
$\left\{ {\left( {1,6} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,6} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,6} \right)} \right\}$
$\left\{ {\left( {1,7} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,6} \right)} \right\}\;\& \,\left\{ {\left( {1,8} \right),\left( {2,7} \right)} \right\}$

8. Originally Posted by Plato
$\left\{ {\left( {1,6} \right),\left( {2,6} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,8} \right)} \right\}$
$\left\{ {\left( {1,6} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,6} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,6} \right)} \right\}$
$\left\{ {\left( {1,7} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,6} \right)} \right\}\;\& \,\left\{ {\left( {1,8} \right),\left( {2,7} \right)} \right\}$
Thank you so much. I keep forgetting that the definition of a function maps every element in the domain to a unique element in the codomain.

This will really help my understanding in the future.