## Prove on equivalence, addition and multplication

let S be the set of ordered pairs of positive integers. Let equivalence, addition and multiplication be defined on the elements of S as follows: For all (a,b) and (c,d) in S: (a,b) is equvalent to (c,d)<-->a+d=b+c; (a,b)+(c,d) = (a+c,b+d) and (a,b)*(c,d)=(ac+bd,ad,bc).
a) Prove that if addens in S are equivalent, then their sums in S are also equivalent.
b)Prove that if factors in S are equivalent, then their products in S are also equivalent.
c)Allowing equivalence, shoe that multiplication is distributive over addition.
d) Show tha [S, +, *] is isomorphic to [Z, +, *] if equicalent elements in S are considered identical and (a,b) in S correspond to a-b in Z.