Cycles, functions that equal their own inverse

[ArcanaNoir]

I'm doing some really basic cycle homework, and in doing an exercise, I noticed something which is what i was probably supposed to notice but I want to know if my conclusion is correct.

"all mappings from N to N (N=finite subset of naturals) that can be written with exclusively 1 or 2 element cycles are equal to their inverse."

For example 1~4, 2~3, 3~2, 4~1 (~ is my quick iPod shorthand for the maps to arrow) is written in cycle notation as (1 4)(2 3)

Can I perhaps extend it to:
"a mapping from N to N (N= finite subset of naturals) is equal to its inverse iff it can be written with exclusively 1 or 2 element cycles."

Can i extend it even more to all finite sets? I've only done cycles on the naturals so far, I don't know if I'm getting way off here. I'm also trying to think about what these functions look like visually. Does anyone know any good continuous functions that are equal to their inverse besides trivial ones like y=x?

 

[I like Serena]

Hi Arcana!

Can I perhaps extend it to:
"a mapping from N to N (N= finite subset of naturals) is equal to its inverse iff it can be written with exclusively 1 or 2 element cycles."

Yes. (edit: with the addition that the cycles are disjoint).
You should try to (dis)proof it yourself though...

Can i extend it even more to all finite sets? I've only done cycles on the naturals so far, I don't know if I'm getting way off here. I'm also trying to think about what these functions look like visually. Does anyone know any good continuous functions that are equal to their inverse besides trivial ones like y=x?

Extend it to all finite sets?
Sure.
That's what isomorphisms have been invented for.

As for continuous functions:
y=1/x is another one...
And y=1-x...

 

[Robert1986]

In this:

"all mappings from N to N (N=finite subset of naturals) that can be written with exclusively 1 or 2 element cycles are equal to their inverse."

I assume that you mean in the map's cycle decomposition. In particular, the two cycles are distinct, e.g. (1,2) and (2,3) can't be cycles, correct? If the transpositions (2-cycles) are not disjoint then what you have written is not true since any mapping from N to N can be written as a bunch of transpositions, but not necessarily distinct.

OK, so to answer your question, the iff thing is correct (at least, it seems to me; I haven't really bothered to prove it a whole bunch.)

 

[ArcanaNoir]

Hi Arcana!
Yes. (edit: with the addition that the cycles are disjoint).
You should try to (dis)proof it yourself though...

Yeah I meant disjoint. It's tedious to check my writing when using my iPod, and I was already wasting time...

Extend it to all finite sets?
Sure.
That's what isomorphisms have been invented for.

Oh really? Cool. I haven't learned what an isomorphism is yet, but I'm sure it's coming soon...

As for continuous functions:
y=1/x is another one...
And y=1-x...

Those are boring/trivial too. Anything cooler? (With a more interesting shape?) Or are they all straight lines?

 

[I like Serena]

Yeah I meant disjoint. It's tedious to check my writing when using my iPod, and I was already wasting time...

Oh that makes it okay then.

Oh really? Cool. I haven't learned what an isomorphism is yet, but I'm sure it's coming soon...

Probably your next chapter.
The word means literally "same shape mapping", which is exactly what it is.

Those are boring/trivial too. Anything cooler? (With a more interesting shape?) Or are they all straight lines?

Don't you think a conic section is a cool shape?
It certainly isn't a straight line.

Don't you have any of your own?
Some creativity would be nice!

What about $\sqrt{1-x^2}$?

 

[ArcanaNoir]

Don't you think a conic section is a cool shape?
It certainly isn't a straight line.

Don't you have any of your own?
Some creativity would be nice!

What about $\sqrt{1-x^2}$?

That's a cooler function for sure. As for my own creativity, I have very little time to devote to diversions such as this. I spend as much time as I can mastering the information that will be on the tests. It's self-indulgent of me to be playing around thusly. However, this last equation does add significant variety to my collection of these functions, and inspires me to think about others, in the pursuit of why they are their own inverse. Interesting... I have lots of thoughts about this! Cycles, finite sets, continuous functions, infinite sets, they seem very unrelated yet somehow they are... What does one property say about another, how does one rule apply to a different notation... Thoughts thoughts thoughts!

I almost don't want to get to the chapter on isomorphisms because it would be like giving away the end of a good story.

 

[I like Serena]

Well, I can't help you with time management.
But I would recommend not to forget that math should be fun, and a few fun diversions are required.

As for isomorphisms, usually they are explained in great detail, what the definition is, how you apply them... but somehow the deeper reason behind them, what they actually mean, is neglected - and it shouldn't!
Because when you get the deeper meaning, the material becomes much easier, because you understand!