Finding a Bijective Correspondence between X^{\omega} and a Proper Subset

[Samuelb88]

Homework Statement

Let X be the two element set \{ 0 , 1 \}. Find a bijective correspondence between X^{\omega} and a proper subset of itself.

Homework Equations

Notation. X^{\omega} is the set of all (infinite) {\omega}-\mathrm{tuples} (x_1 , x_2 , x_3 , ...), where x_i \in X.

The Attempt at a Solution

My question is about the proper subset part...

I want to say in order to find any such bijection, I'll need to find another infinite proper subset of X^{\omega}. My question is, does X^{\omega - r}, where r \in \mathbb{N}, constitute such a proper subset?

 

[vela]

What would you suggest X^{\omega-r} means? How would it differ from X^\omega?

It might help to think about the binary representation of numbers.

 

[micromass]

To help you, consider the set of all tuples

(0,x_2,x_3,x_4,...)

so the set of all tuples with 0 as first element. Try to use that set somewhere...

 

[Samuelb88]

That makes sense! I let \alpha = \{ A \in \{0,1\}^{\omega} : A = (0, x_1, x_2 , ...) \} and define f: \{0,1\}^{\omega} \rightarrow \alpha such that f(x_1, x_2, ...) = (0, x_1, x_2, ...), that is, the function that shifts each coordinate position of any \omega-tuple in \{0,1\}^{\omega} "up by one" to compensate for the zero in the first coordinate position after I put it through my function. Thanks!