Chapter A, Problem 21E

Define a relation on the set of ordered pairs of positive integers by $\left(w,x\right)~\left(y,z\right)$ if and only if $w+z=x+y$ . Show that the operations ${\left[\left(w,x\right)\right]}_{~}+{\left[\left(y,z\right)\right]}_{~}={\left[\left(w+y,x+z\right)\right]}_{~}$ and ${\left[\left(w,x\right)\right]}_{~}\cdot {\left[\left(y,z\right)\right]}_{~}={\left[\left(w\cdot y+x\cdot z,x\cdot y+w\cdot z\right)\right]}_{~}$

1. are well-defined, that is, they do not depend on the representative of the equivalence classes chosen for the computation.