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Proove countability

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  • #1
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Homework Statement


Proove that Nx{0} is countable. x stand for a product i.e. like the cartesian product NxN


Homework Equations


N is countable.


The Attempt at a Solution


This is so obvious since Nx{0} is just (1,0),(2,0) etc. But how do you write a proof formally?
 

Answers and Replies

  • #2
Dick
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Homework Statement


Proove that Nx{0} is countable. x stand for a product i.e. like the cartesian product NxN


Homework Equations


N is countable.


The Attempt at a Solution


This is so obvious since Nx{0} is just (1,0),(2,0) etc. But how do you write a proof formally?
Show there is a 1-1 correspondence between Nx{0} and N. Write an explicit bijection. Yes, it is easy.
 
  • #3
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So I can define the map:
f: N->Nx{0}
f(x)=(x,0)
and state that this is clearly bijective as a proof?
 
  • #4
jbunniii
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So I can define the map:
f: N->Nx{0}
f(x)=(x,0)
and state that this is clearly bijective as a proof?
I suggest writing the extra few lines to prove injectivity and surjectivity instead of writing "clearly." The problem statement itself seems so obvious that it's tempting to write "clearly" as a one-word proof, but clearly that isn't the intent.
 

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