# Math Help - Abstract questions

1. ## Abstract questions

1)Suppose a1, a2,.....,an are denumerable sets. Show that a1Xa2X....Xan is denumberable(a1Xa2X...xan is the set of n-tuples(a1,a2...an),ai is an elember of Ai)

I literally have no idea what that means so any direction would be nice(a1 is a sub 1 btw)

2)a)Suppose 0<=a<b and o<=c<d, prove that 0<=ac<bd

b)0<=a<b then 0<=a^2<b^2

c)if 0<a<b, show that a<rt(ab)<(a+b)/2<b (hint contradiction)

2. ## Re: Abstract questions

Originally Posted by Johngalt13
1)Suppose a1, a2,.....,an are denumerable sets. Show that a1Xa2X....Xan is denumberable(a1Xa2X...xan is the set of n-tuples(a1,a2...an),ai is an elember of Ai)

I literally have no idea what that means so any direction would be nice(a1 is a sub 1 btw)

2)a)Suppose 0<=a<b and o<=c<d, prove that 0<=ac<bd

b)0<=a<b then 0<=a^2<b^2

c)if 0<a<b, show that a<rt(ab)<(a+b)/2<b (hint contradiction)

1) You need to prove it for $n=2$ and the rest will follow by induction. To do this note that $\mathbb{N}^2\to\mathbb{N}: (a,b)\mapsto 2^a3^b$ is an injection, and since there is a natural injection $\mathbb{N}\to\mathbb{N}^2$ the rest follows from the Schroeder-Bernstein theorem. Use this then to show that $A,B\simeq\mathbb{N}$ implies $A\times B\simeq\mathbb{N}$ by creating mappings of the form $A\times B\leftrightarrow \mathbb{N}^2\leftrightarrow\mathbb{N}$.

3. ## Re: Abstract questions

Ah thanks a lot, I got that one now. Any idea how to start the second one? I'm working on it now and just keep going in circles.

4. ## Re: Abstract questions

Originally Posted by Johngalt13
Ah thanks a lot, I got that one now. Any idea how to start the second one? I'm working on it now and just keep going in circles.
It all depends upon where you are starting. If we are working in some ordered ring (e.g. what is in Rudin) this is true by definition.

5. ## Re: Abstract questions

Edit: Nevermind, I figured it out, thanks though.