Final answer:
To show R is an equivalence relation, it must be proven that R is reflexive, symmetric, and transitive. By demonstrating R satisfies these three properties for the given conditions on ordered pairs of integers, it is confirmed that R is indeed an equivalence relation.
Step-by-step explanation:
To show that a relation R is an equivalence relation, we have to prove that it is reflexive, symmetric, and transitive.
Reflexive Property
For any ordered pair of integers (x, y) in the set A, (x, y) R (x, y) because xy = yx which is true. So, the relation R is reflexive.
Symmetric Property
If (x, y) R (u, v), then xv = yu. To prove symmetry, we need to show that (u, v) R (x, y) also holds. Since xv = yu, we also have yu = xv, which shows that (u, v) R (x, y), fulfilling the symmetric property.
Transitive Property
Suppose (x, y) R (u, v) and (u, v) R (w, z), which means xv = yu and uz = vw. Multiplying these two equations, we get xvyz = yuwv. Because v ≠ 0, we can simplify it to xz = yw, which shows (x, y) R (w, z). This proves R is transitive.
Since R has been shown to have the reflexive, symmetric, and transitive properties, it is an equivalence relation on the set A of ordered pair of integers.