Math Is Fun Forum

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Linear Algebra Help

I'm not quite sure how to answer this question:

Let X and Y be sets with

and

.

Determine the number of functions f mapping X into Y.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Linear Algebra Help

With a mapping, we can't send an x to more than one y.  So how many places are there for you to send x?  And how many x's do you have to send?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Linear Algebra Help

That's the thing I'm not too sure.  I thought it had something to do with the values n and m.  I assume n suggests that there are n values in the set X and m values in set Y, then from there we find out how many functions could be mapped.  Does that help or was that load of b*ll**ks!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Linear Algebra Help

Yes, that you definitely need to know.

A map is determined by where things get sent.  Once you know how many places you can send things, you can pretty easily figure out how many maps their are.  So how many places can you send an arbitrary element x in X?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Linear Algebra Help

I understand what you are saying about mapping, but I'm afraid that's all there is to that question.

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Linear Algebra Help

Anyone any ideas?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Linear Algebra Help

Sometimes the best explanation of a problem is the answer itself. For this problem, it’s

There are m[sup]n[/sup] possible functions from a set with n elements to a set with m elements. That’s the answer; it’s your job now to reason out why it is so.
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Last edited by JaneFairfax (2008-10-02 02:36:01)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Linear Algebra Help

I have asked twice now:

So how many places can you send an arbitrary element x in X?

If you can't answer that after some thought, then I suggest you go back over what the definition of a function is, in terms of relations.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Linear Algebra Help

I think JaneFairfax is going about this the right way because Ricky I don't think I can give you an answer to that question you are asking except that there are n elements in X that can be mapped to m elements in Y.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Linear Algebra Help

I don't think I can give you an answer to that question you are asking except that there are n elements in X that can be mapped to m elements in Y.

It seems you don't understand the question, but you do understand the answer.  For how many places a single element in X can be sent, it does not matter how many elements there are in X.  That has absolutely no bearing on the number of places that it can get sent.  However, the number that is important here is the one you said: m.  There are m places to send any element in X.

Now you have the answer for a single element of X.  Now the next step: You can send an arbitrary element, call it x1, in X to m places.  Now lets pick another element, x2, with x1 not equal to x2.  How many places can x2 go?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

shriya

Guest

Re: Linear Algebra Help

i am not able to do this question as it is so difficult pleas help me....:(