# Understanding Permutations

Hello,

I am having trouble understanding permutations. Here is the definition that I was given:

Let A be a nonempty set, A permutation of A is any function [alpha]: A --> A such that [alpha] is both one-to-one and onto.

Then the example:

Let A = {1,2}. Then the permutations of A are

1[alpha] = 1
2[alpha] = 2
1[beta] = 2
2[beta] = 1

Now I know this is basic, basic stuff. And many of you will say "What the heck?!!? Just think about it!" But I have thought about it. And for a long time. I just have trouble with some really elementry subjects sometimes. I was hoping if someone just rephrases it for me maybe I will understand it a bit better. I am mostly having trouble understanding the example. But perhaps I am not understanding the definition as well. For if I did I would understand the definition. [?]

Here is what I know about one-to-one, onto functions.

I know that for a function to be one-to-one, every x that is an element of the set A, there should be an x[alpha] which is also an element in A.

And for a function to be onto, for every x[alpha] that is in A, there should be an x which is an element of A.

(PLEASE correct me if my definitions are incorrect. OR elaborate on the definitions if you deem them insufficient.)

Now on an intuitive level, I "THINK" I know what a permutation is - any possible combo of the elements in A. NOTE: I didn't say that my interpretation is correct.

Now, for the first two parts of the example

1[alpha]=1
2[alpha]=2

I interpret this to be some function such as sort of an identity function.

With the second two parts of the example, I SEE what is going on, but I can't explain it. I see that put an input x into a function and get y, put y into the same function and get x .

If anyone has the time or patience to SPOONFEED me on this definition I would appreciate it greatly.

Note: I am also aware that given n elements of a set there are n! combinations of these elements.

Sincerely,

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