222k views
4 votes
Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.

User Sdornan
by
7.5k points

1 Answer

3 votes

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.

User Maniganda Prakash
by
9.0k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories