Final answer:
The relation R on the set A of ordered pairs of non-zero integers, defined by (x, y)R(u, v) if xv = yu, is shown to be an equivalence relation by verifying the three necessary properties: reflexivity, symmetry, and transitivity.
Step-by-step explanation:
To demonstrate that the given relation R on the set A of ordered pairs of non-zero integers is an equivalence relation, we must show that it satisfies three properties: reflexivity, symmetry, and transitivity. The relation R is defined by (x, y)R(u, v) if and only if xv = yu.
Reflexivity
For any pair (x, y) in A, we consider the product xy, which is always equal to yx. Therefore, (x, y)R(x, y) because xy = yx, satisfying the reflexivity condition.
Symmetry
If we have two pairs (x, y) and (u, v) such that (x, y)R(u, v), this means xv = yu. If we reverse the roles of these pairs, we get vu = uy, which also implies (u, v)R(x, y), satisfying the symmetry condition.
Transitivity
Consider three pairs (x, y), (u, v), and (s, t) such that (x, y)R(u, v) and (u, v)R(s, t), meaning xv = yu and ut = vs. Multiplying these two equations, we get xvtu = yust. Since the non-zero integers form an integral domain, we can cancel the common factors to get xt = ys, showing that (x, y)R(s, t), satisfying the transitivity condition.
Therefore, considering reflexivity, symmetry, and transitivity, we can conclude that the relation R is an equivalence relation on the set A of ordered pairs of non-zero integers.