Manifold
September 18th, 2004, 5:03 AM
I've got an exercise I would like to discuss with you...I came to this idea because of the thread "even and odd numbers" which has a lot to do with it...
Task (Source: V.A. Zorich, Mathematical Analysis 1, Springer-Verlag):
a) Prove the equipollence of the closed interval \{x\in\mathbb{R}~|~0\le{x}\le{1}\} and the open interval \{x\in\mathbb{R}~|~0<x<1\} of the real line \mathbb{R} both using the Schröder-Bernstein theorem and by direct exhibition of a suitable bijection.
b) Analyze the following proof of the Schröder-Bernstein theorem:
(card~X\le{card~Y})\wedge(card~Y\le{card~X}) \Rightarrow (card~X=card~Y).
Proof:
It suffices to prove that if the sets X,Y,Z are such that X\supset{Y}\supset{Z} and card~X=card~Z, then card~X=card~Y. Let f:X\rightarrow{Z} be a bijection. A bijection g:X\rightarrow{Y}can be defined, for example, as follows:
g(x)=\left\{{f(x),~if~x\in{f^n(X)\setminus{f^n(Y)} ~for~some~n\in\mathbb{N},}\atop~{x,~otherwise.}\
Here f^n=f\circ{...}\circ{f} is the nth iteration of the mapping f and \mathbb{N} is the set of natural numbers. (Remark: N={1,2,3,...} in this terminology)
Task (Source: V.A. Zorich, Mathematical Analysis 1, Springer-Verlag):
a) Prove the equipollence of the closed interval \{x\in\mathbb{R}~|~0\le{x}\le{1}\} and the open interval \{x\in\mathbb{R}~|~0<x<1\} of the real line \mathbb{R} both using the Schröder-Bernstein theorem and by direct exhibition of a suitable bijection.
b) Analyze the following proof of the Schröder-Bernstein theorem:
(card~X\le{card~Y})\wedge(card~Y\le{card~X}) \Rightarrow (card~X=card~Y).
Proof:
It suffices to prove that if the sets X,Y,Z are such that X\supset{Y}\supset{Z} and card~X=card~Z, then card~X=card~Y. Let f:X\rightarrow{Z} be a bijection. A bijection g:X\rightarrow{Y}can be defined, for example, as follows:
g(x)=\left\{{f(x),~if~x\in{f^n(X)\setminus{f^n(Y)} ~for~some~n\in\mathbb{N},}\atop~{x,~otherwise.}\
Here f^n=f\circ{...}\circ{f} is the nth iteration of the mapping f and \mathbb{N} is the set of natural numbers. (Remark: N={1,2,3,...} in this terminology)