Bijective Mapping of Cartesian Products: X^{m} \times X^{n} to X^{m + n}?

[QuantumP7]

Homework Statement

Find a bijective map g : X^{m} \times X^{n} \rightarrow X^{m + n}

Homework Equations

The Attempt at a Solution

I don't even know where to begin. How would I map X^{m} \times X^{n} in the first place? How could I map X^{m + n}?

 

[Dick]

It's a simple question. The only reason I can see that you would have a problem is that you don't know what the terms mean. What does X^m mean? What does (X^m)x(X^n) mean? Start looking up the definitions and state them here and I'll try and help.

 

[QuantumP7]

It's a simple question. The only reason I can see that you would have a problem is that you don't know what the terms mean. What does X^m mean? What does (X^m)x(X^n) mean? Start looking up the definitions and state them here and I'll try and help.

I think that (X^{m}) \times (X^{n}) means the cartesian product between the m-tuples of the set X, and the n-tuples of X. So it would be (x_{1}, x_{2}, \cdots, x_{m}) and (x_{1}, x_{2}, \cdots, x_{n})? So the cartesian product of the two would be (if m < n) (x_{1}, x_{2}, \cdots, x_{m}, x_{m + 1}, \cdots, x_{n})?

 

[Dick]

I think that (X^{m}) \times (X^{n}) means the cartesian product between the m-tuples of the set X, and the n-tuples of X. So it would be (x_{1}, x_{2}, \cdots, x_{m}) and (x_{1}, x_{2}, \cdots, x_{n})? So the cartesian product of the two would be (if m < n) (x_{1}, x_{2}, \cdots, x_{m}, x_{m + 1}, \cdots, x_{n})?

Good so far. So (X^m)x(X^n) is all of the (m+n)-tuples like (x1,x2,...xm,x_m+1,...,x_m+n). You want to split that into an m-tuple (in X^m) and an n-tuple (in X^n). How about making the m-tuple (x1...xm) and the n-tuple (x_m+1...x+_m+n)? Does that define a bijection? I think it does. It's not the only one, but you only need one. Do you see why it works?

 

[QuantumP7]

That makes perfect sense! Thank you so much!

So, the X^{m + n} would just be the X^{m} and X^{n} together? If so, I can definitely see a bijection between them!

 

[Dick]

That makes perfect sense! Thank you so much!

So, the X^{m + n} would just be the X^{m} and X^{n} together? If so, I can definitely see a bijection between them!

Yes, pretty much. You might want to think about how you would formally prove it's a bijection if needed. But seeing how it works is good start.

 

[QuantumP7]

Ok, I get it! Thank you soooooo much!