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?
Being countable means that the set has a finite number of elements or can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, ...).
To prove that Nx{0} is countable, we need to show that there exists a bijection (a one-to-one and onto function) between Nx{0} and the set of natural numbers. This can be done by constructing a function that maps each element in Nx{0} to a unique natural number, and showing that every natural number is mapped to by at least one element in Nx{0}.
Proving that Nx{0} is countable is important because it helps us understand the cardinality (size) of this set. It also allows us to make comparisons between Nx{0} and other sets, and to determine if they have the same cardinality or not.
One example of a bijection between Nx{0} and the set of natural numbers is the function f(n) = n, where n is a natural number. This function maps each element in Nx{0} to a unique natural number, and every natural number is mapped to by at least one element in Nx{0}.
Yes, a set can be infinite and still be countable. As long as there exists a bijection between the set and the set of natural numbers, it is considered to be countable. This means that even though the set may have an infinite number of elements, they can still be put into a one-to-one correspondence with the natural numbers.