Math Is Fun Forum

Identity

Member

Registered: 2007-04-18

Posts: 934

bijective inverse function

Let

be one-one and onto. Then show that the inverse function

is one-one and onto.

Could someone please help me with writing out a rigorous proof of this? I need practice writing proofs and I'm not sure how to set it out
Thanks

Last edited by Identity (2010-03-05 03:34:02)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: bijective inverse function

It would be much more beneficial if you wrote what you thought was rigorous, and let us critique it.

---

"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..."

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: bijective inverse function

Proof of one-to-one-ness:

Let

and let

Since

(this step seems a bit flimsy to me... it's just what I'm used to... am I taking it for granted?)

and

So

is one-to-one.

Converse:

If

then

?? Not sure about the converse...

Proof of onto-ness:

If

is one-to-one if for every

there is a

s.t.

.

err... not sure where to go from here

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: bijective inverse function

Your proof of the injectiveness of

is perfect.

There is no necessity to prove the converse, which is merely establishing that the inverse function is well defined. Still, if you want to prove it, use the fact that

is injective.

To prove ontoness, take an

and find a

such that

. The obvious choice would be

.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: bijective inverse function

Thanks Jane!

Proof of onto-ness:

Ok, so
"

is onto if for every

s.t.

"

Choose

Then we have

is onto!

Is that all there is to it?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: bijective inverse function

Yes, that’s all there is to it.