Define f: ℕ×ℕ→ℕ as follows: For each (m,n) ∈ ℕ×ℕ, $\displaystyle f(m,n) =2^{m-1}*(2n-1) $
Prove that f is an injection.
I don't know how to go about doing this one.
Suppose that $\displaystyle f(m,n)=f(m',n')$, then we'd have that $\displaystyle 2^{m-1}(2n-1)=2^{m'-1}(2n'-1)$.
So let me ask you this, we can assume WLOG that $\displaystyle m\leqslant m'$, right? So this means that $\displaystyle m'-1-(m-1)=m'-m\geqslant 0$ and so dividing through we get $\displaystyle 2^{m'-m}(2n-1)=2n'-1$. Now, if $\displaystyle m'-m>0$ what would the problem be with that?Spoiler: