Question

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.

Answer 1

Answer:

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.