Math Is Fun Forum

morik

Member

Registered: 2005-11-23

Posts: 16

Sets, Functions, Injunctions and Bijunctions

I have been given this question to answer as either integer numbers or formulas...

Given two sets A and B with |A| = 9 and |B| = 17.

1. What is the number of functions from A to B?

2. What is the number of injective (one-to-one) functions from A to B?

3. What is the number of bijective (one-to-one and onto) functions from A to A?

Any explanations and help would be much appreciated. Thanks!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Sets, Functions, Injunctions and Bijunctions

Set theory combined with combinatorics, I love these kind of questions!

So how does one go about creating a finite mapping?  Pretty simple actually.  For each element in set A, choose an element in set B without restriction.  How many choices do you have for one element in A and how many elements are in A?

How do you create a finite mapping that is 1-1?  Again, for each element of A, choose an element in B.  But, once an element is taken, you may not choose this element again.

Finally, bijective.  Perhaps unexpectedly, this is the same answer as the question above.  But, as one progresses, you find that between two finite sets of the same size, every 1-1 mapping is also onto.  The proof of this is fairly easy.

---

"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..."