Cardinality of infinite sets

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  • #1
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Homework Statement


Prove that if A,B, and C are nonempty sets such that A [itex]\subseteq[/itex] B [itex]\subseteq[/itex] C and |A|=|C|, then |A|=|B|

The Attempt at a Solution


Assume B [itex]\subset[/itex] C and A [itex]\subset[/itex] B (else A=B or B=C), and there must be a bijection f:A[itex]\rightarrow[/itex]C...
 

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  • #2
Dick
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Homework Statement


Prove that if A,B, and C are nonempty sets such that A [itex]\subseteq[/itex] B [itex]\subseteq[/itex] C and |A|=|C|, then |A|=|B|

The Attempt at a Solution


Assume B [itex]\subset[/itex] C and A [itex]\subset[/itex] B (else A=B or B=C), and there must be a bijection f:A[itex]\rightarrow[/itex]C...
So far you are just stating what the problem told you. Don't you have some theorems you might apply?
 
  • #3
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Homework Statement


Prove that if A,B, and C are nonempty sets such that A [itex]\subseteq[/itex] B [itex]\subseteq[/itex] C and |A|=|C|, then |A|=|B|

The Attempt at a Solution


Assume B [itex]\subset[/itex] C and A [itex]\subset[/itex] B (else A=B or B=C), and there must be a bijection f:A[itex]\rightarrow[/itex]C...
Why not start with the given condition, that A [itex]\subseteq[/itex] B [itex]\subseteq[/itex] C and |A|=|C|?
 
  • #4
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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?
 
  • #5
Dick
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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?
 
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  • #6
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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?
 
  • #7
Dick
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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.
 

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