# Cardinality of infinite sets

1. Nov 27, 2012

### rideabike

1. The problem statement, all variables and given/known data
Prove that if A,B, and C are nonempty sets such that A $\subseteq$ B $\subseteq$ C and |A|=|C|, then |A|=|B|

3. The attempt at a solution
Assume B $\subset$ C and A $\subset$ B (else A=B or B=C), and there must be a bijection f:A$\rightarrow$C...

2. Nov 27, 2012

### Dick

So far you are just stating what the problem told you. Don't you have some theorems you might apply?

3. Nov 27, 2012

### Staff: Mentor

Why not start with the given condition, that A $\subseteq$ B $\subseteq$ C and |A|=|C|?

4. Nov 27, 2012

### rideabike

I know, I don't really know where to start. Schroder-Bernstein maybe?

5. Nov 27, 2012

### Dick

That's the one! Try and apply it. Here's a hint. If A is subset of B, then there's an injection from A into B, right?

Last edited: Nov 27, 2012
6. Nov 27, 2012

### rideabike

Right. And we want to show there's an injection from B to A. Would it be that since there's an injection from B to C and and injection from C to A, there must be an injection from B to A?

7. Nov 27, 2012

### Dick

Sure. That's wasn't so hard, was it? You might want to spell out some of the details, like what the actual injections are in terms of your bijection f:A->C. But that's the idea.