Does Equal Cardinality in Nested Infinite Sets Imply Equality Throughout?

[rideabike]

Homework Statement

Prove that if A,B, and C are nonempty sets such that A $\subseteq$ B $\subseteq$ C and |A|=|C|, then |A|=|B|

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...

 

[Dick]

Homework Statement

Prove that if A,B, and C are nonempty sets such that A $\subseteq$ B $\subseteq$ C and |A|=|C|, then |A|=|B|

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...

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

 

[Mark44]

Homework Statement

Prove that if A,B, and C are nonempty sets such that A $\subseteq$ B $\subseteq$ C and |A|=|C|, then |A|=|B|

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...

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

 

[rideabike]

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

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

 

[Dick]

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

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?

 

[rideabike]

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?

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?

 

[Dick]

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?

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.