What Is the Meaning of Permutations in Mathematics?

[wubie]

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.

 

[wubie]

Alright. Nevermind my garbage up there. I finally figured it out. It only took about five to six hours but I was able to get it.

Thanks to all who took the time to browse my post.

Cheers.

 

[M98Ranger]

Hello,

I completely understand your confusion with permutations. It can be a tricky concept to grasp at first, but once you understand it, it becomes much easier to work with. Let me try to explain it in a different way that may help you understand it better.

A permutation is essentially just a rearrangement of the elements in a set. Think of it like shuffling a deck of cards. You have a set of cards, and you can rearrange them in any order you want. In the same way, a permutation is a function that rearranges the elements in a set.

Now, let's break down the definition. A permutation is a function that takes elements from a set A and maps them back to the same set A. This means that the output of the function must be an element of the same set A. In other words, the elements in the set A are being rearranged, but they are still the same elements.

Next, the function must be both one-to-one and onto. One-to-one means that for every input, there is only one corresponding output. In the example, 1[alpha] = 1 and 2[alpha] = 2, which means that every input (1 and 2) has a unique output (also 1 and 2). This is important because it ensures that no elements are repeated or skipped in the rearrangement.

Onto means that every element in the set A must have a corresponding output. In the example, both 1[beta] and 2[beta] map to each other, meaning that all elements in the set A (1 and 2) have a corresponding output. This is important because it ensures that no elements are left out in the rearrangement.

Now, for the example itself. Let's look at the first two parts: 1[alpha] = 1 and 2[alpha] = 2. This is just like the identity function you mentioned. This function is essentially saying "take the elements in the set A and leave them in their original order." So, 1 stays 1 and 2 stays 2.

The last two parts, 1[beta] = 2 and 2[beta] = 1, are where the rearranging happens. This function is saying "take 1 and switch it with 2, and take 2 and switch it with 1." So, the elements in the set A are