The discussion revolves around proving that the set \( \mathbb{N} \times \{0\} \) is countable, with participants exploring the nature of this set as a Cartesian product involving natural numbers.
Some participants have offered guidance on the need for a more formal proof, suggesting that additional lines proving injectivity and surjectivity would enhance the argument. There is recognition of the temptation to simplify the proof due to the perceived obviousness of the problem.
Participants note the challenge of balancing the simplicity of the problem with the requirement for a rigorous proof format, indicating a tension between intuitive understanding and formal mathematical standards.
aaaa202 said: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?
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.aaaa202 said:So I can define the map:
f: N->Nx{0}
f(x)=(x,0)
and state that this is clearly bijective as a proof?