Answer :

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


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