# Cardinality of infinite sets

## 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...

## Answers and Replies

Related Calculus and Beyond Homework Help News on Phys.org
Dick
Homework Helper

## 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
Mentor

## 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|?

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
Homework Helper
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?

Last edited:
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