function composition and inverse function

**sbankica** · November 9th 2009, 01:03 PM

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}?

**Plato** · November 9th 2009, 01:29 PM

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?

**sbankica** · November 9th 2009, 01:33 PM

4^3?

**sbankica** · November 9th 2009, 01:34 PM

so answer is: (3^2)(4^3)

**Plato** · November 9th 2009, 01:41 PM

Originally Posted by sbankica

so answer is: (3^2)(4^3)

You got it!

**oldguynewstudent** · November 9th 2009, 08:13 PM

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.

**Plato** · November 10th 2009, 03:07 AM

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?

$\left\{ {\left( {1,6} \right),\left( {2,6} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,8} \right)} \right\}$
$\left\{ {\left( {1,6} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,6} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,6} \right)} \right\}$
$\left\{ {\left( {1,7} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,6} \right)} \right\}\;\& \,\left\{ {\left( {1,8} \right),\left( {2,7} \right)} \right\}$

**oldguynewstudent** · November 10th 2009, 12:18 PM

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.

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