Show that the relation S defined on the set N × N by

(a, b) S (c, d) ⇒ a + d = b + c is a equivalence relation

Given: (a, b) S (c, d) ⇒ a + d = b + c

To prove: given relation is equivalence relation

A relation is said to be an equivalence relation if it is reflexive, symmetric and transitive

Step 1:

Now, (a, b) S (b, a)

⇒ a + b = b + a which is true

∴ Relation R is reflexive

Step 2:

Now, (a, b) S (c, d)

⇒ a + b = c + d

⇒ c + d = a + b

⇒ (c, d) S (a, b) which is true

∴ Relation R is symmetric

Step 3:

Now, (a, b) S (c, d) and (c, d) S (e, f)

⇒ a + b = c + d and c + d = e + f

⇒ a + b = e + f

⇒ (a, b) S (e, f) which is true

∴ Relation R is transitive

This shows that Relation S is an equivalence relation