Chapter 1.2, Problem 13E

For the given $f:A\to B,$ decide whether $f$ is onto and whether it is one-to-one. Prove that yourdecisions are correct.

a. ${\rm A}=Z\times Z,B=Z\times Z,f\left(x,y\right)=\left(y,x\right)$

b. ${\rm A}=Z\times Z,B=Z,f\left(x,y\right)=x+y$

c. ${\rm A}=Z\times Z,B=Z,f\left(x,y\right)=x$

d. ${\rm A}=Z,B=Z\times Z,f\left(x\right)=\left(x,1\right)$

e. ${\rm A}=Z\times Z,B=Z,f\left(x,y\right)=x^2$

f. ${\rm A}=Z\times Z,B=Z,f\left(x,y\right)=x^2+y^2$

g. ${\rm A}=Z^{+}\times Z^{+},B=Q,f\left(x,y\right)=x/y$

h. ${\rm A}=R\times R,B=R,f\left(x,y\right)=2^{x+y}$