Thread: Help with a problem

Simple problem from Amazon.com: Discovering Modern Set Theory. I: The Basics (Graduate Studies in Mathematics, Vol…

We define a function G from [0, 1) x [0, 1) to [0, 1) as follows:

For any <x, y> in [0, 1) x [0, 1) such that x = 0.x1x2x3x4x5... and y = y1y2y3y4y5... G(x, y) = 0.x1y1x2y2x3y3....

Clearly this is into. However, in the book they claim it is not onto. I can't see why this is. My reasoning is:

Take any z in [0, 1) such that z = 0.z1z2z3z4z5.... then x = 0.z1z3z5z7... and y = 0.z2z4z6z8... will be such that G(x, y) = z.

Any help would be great.

Regards
Sam

Originally Posted by sroberts

Simple problem from Amazon.com: Discovering Modern Set Theory. I: The Basics (Graduate Studies in Mathematics, Vol…

We define a function G from [0, 1) x [0, 1) to [0, 1) as follows:

For any <x, y> in [0, 1) x [0, 1) such that x = 0.x1x2x3x4x5... and y = y1y2y3y4y5... G(x, y) = 0.x1y1x2y2x3y3....

Clearly this is into. However, in the book they claim it is not onto. I can't see why this is. My reasoning is:

Take any z in [0, 1) such that z = 0.z1z2z3z4z5.... then x = 0.z1z3z5z7... and y = 0.z2z4z6z8... will be such that G(x, y) = z.

Any help would be great.

Regards
Sam

EDIT: Oh, those are supposed to be subscripts! Nevermind what I wrote before.

Hi!

Yea, sorry about that; I'm Latex intolerant!

The problem is also here on page 35 Discovering Modern Set Theory: The ... - Google Books

Thanks

Sam

Originally Posted by sroberts

Hi!

Yea, sorry about that; I'm Latex intolerant!

The problem is also here on page 35 Discovering Modern Set Theory: The ... - Google Books

Thanks

Sam

Well it's obvious what you meant now; I just didn't have the right pair of eyes on. I believe the issue is that 0.999999... = 1, and that you can choose a z for instance 0.9090909090... where your proposal would give (x,y) = (1,0) which is not in the domain.

That's great!

Thanks ever so much. I was thinking in this area at one point, but had forgotten that .9999999.. = 1.

Sam

Originally Posted by sroberts

That's great!

Thanks ever so much. I was thinking in this area at one point, but had forgotten that .9999999.. = 1.

Sam

You're welcome! But after some thought, I have reservations about whether the function is actually a function.

Consider (x,y) = (0.19999..., 0.19999...) = (0.20000..., 0.20000...).

Applying G to the first yields G(x,y) = 0.119999... = 0.12

Applying G to the second yields G(x,y) = 0.220000... = 0.22

Also, G(0.19999..., 0.20000...) = 0.12909090... which is again different.

Thoughts?

EDIT: Oh I see this is mentioned at the transition between page 34 and page 35. So the function is a valid function, and the thing about z = 0.90909090... still holds.

That's right.

Thanks again though.

Sam