Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈ R if and only if a + d = b + c. Show that R is an equivalence relation.

Therefore, we conclude that R is an equivalence relation.

Step-by-step explanation:

We know that a relation on a set is called an equivalence relation if it is reflexive, symmetric, and transitive.

R is refleksive because we have that a+b = a+b.

R is symmetric because we have that a+d = b+c equivalent with b+c = a+d.

R is transitive because we have that:

((a, b), (c, d)) ∈ R; ((c, d), (e, f)) ∈ R

a+d = b+c ⇒ a-b=c-d

c+f = d+e ⇒ c-d = e-f

we get

a-b=e-f ⇒ a+f=b+e ⇒ ((a, b), (e, f)) ∈ R.

Therefore, we conclude that R is an equivalence relation.