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.
Yea, sorry about that; I'm Latex intolerant!
The problem is also here on page 35 Discovering Modern Set Theory: The ... - Google Books
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.
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.