Let Z be the set of all integers. Consider the relation R on the Cartesian product Z×Z defined by
(a,b)R(c,d) if and only if a+d=b+c.
a. (i) Prove that R is reflexive, symmetric and transitive on Z×Z.
b. (ii) Hence, justify that R is an equivalence relation on Z×Z.
c. (iii) Determine the equivalence class of the ordered pair (3,5).
d. (iv) Describe in set-builder form the distinct equivalence classes generated by R.