# Abstract questions

• Nov 27th 2011, 09:01 PM
Johngalt13
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)
• Nov 27th 2011, 09:39 PM
Drexel28
Re: Abstract questions
Quote:

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 $A\times B\leftrightarrow \mathbb{N}^2\leftrightarrow\mathbb{N}$.
• Nov 27th 2011, 10:29 PM
Johngalt13
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.
• Nov 27th 2011, 10:40 PM
Drexel28
Re: Abstract questions
Quote:

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.
• Nov 27th 2011, 10:47 PM
Johngalt13
Re: Abstract questions
Edit: Nevermind, I figured it out, thanks though.