Define f: ℕ×ℕ→ℕ as follows: For each (m,n) ∈ ℕ×ℕ,
Prove that f is an injection.
I don't know how to go about doing this one.
Suppose that , then we'd have that .
So let me ask you this, we can assume WLOG that , right? So this means that and so dividing through we get . Now, if what would the problem be with that?
So let me ask you this, we can assume WLOG that , right? So this means that and so dividing through we get . Now, if what would the problem be with that?
Spoiler:
think parity
I don't know or understand what the problem would be.
The left is even and the right is odd but what does that have to with m'-m>0?
If it is true the LHS is even and the RHS is odd, right? But, as we said that's ridiculous. So it can't be true that and since for sure and it's an integer what may we conclude?
If it is true the LHS is even and the RHS is odd, right? But, as we said that's ridiculous. So it can't be true that and since for sure and it's an integer what may we conclude?
Well if it equals zero, the left is odd and the right is odd. If it is >0, the right is even and the left is odd.