Chapter 3.1, Problem 22E

a)

To determine

Whether the function $f\left(n,m\right)=2n+3m;f:\text{N}\times \text{N}\to \text{N}$ is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

b)

To determine

Whether the function $f\left(n,m\right)=2n+3m;f:\text{Z}\times \text{Z}\to \text{Z}$ is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

c)

To determine

Whether the function $f\left(n,m\right)=14n+22m;f:\text{N}\times \text{N}\to \text{N}$ is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

d)

To determine

Whether the function $f\left(n,m\right)=89n+246m;f:\text{Z}\times \text{Z}\to \text{Z}$ is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

e)

To determine

Whether the function $f\left(n,m\right)=n^2+m^2+1;f:\text{Z}\times \text{Z}\to \text{N}$ is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

f)

To determine

Whether the function $f\left(n,m\right)=\lfloor \frac{n}{m}\rfloor +1;f:\text{N}\times \text{N}\to \text{N}$ is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.