Solution:
First let R be a relation on A
It is given that set A of ordered pair of integers defined by (x, y) R (u, v) if xv = y u
Now we have to check whether the given relation is equivalence or not.
To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.
Reflexivity:
Let (a, b) be an arbitrary element of the set A.
Then, (a, b) ∈ A
⇒ a b = b a
⇒ (a, b) R (a, b)
Thus, R is reflexive on A.
Symmetry:
Let (x, y) and (u, v) ∈A such that (x, y) R (u, v). Then,
x v = y u
⇒ v x = u y
⇒ u y = v x
⇒ (u, v) R (x, y)
So, R is symmetric on A.
Transitivity:
Let (x, y), (u, v) and (p, q) ∈R such that (x, y) R (u, v) and (u, v) R (p, q)
⇒ x v = y u and u q = v p
Multiplying the corresponding sides, we get
x v × u q = y u × v p
⇒ x q = y p
⇒ (x, y) R (p, q)
So, R is transitive on A.
Therefore R is reflexive, symmetric and transitive.
Hence, R is an equivalence relation on A.