Thread: 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)

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 and the rest will follow by induction. To do this note that is an injection, and since there is a natural injection the rest follows from the Schroeder-Bernstein theorem. Use this then to show that implies by creating mappings of the form .

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.

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.

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