Q. 145.0( 1 Vote )

Mark the tick against the correct answer in the following:

Let R be a relation on N × N, defined by
(a, b) R (c, d)
a + d = b + c. Then, R is


A. reflexive and symmetric but not transitive

B. reflexive and transitive but not symmetric

C. symmetric and transitive but not reflexive

D. an equivalence relation

Answer :

According to the question ,


R = {(a, b) , (c, d) : a + d = b + c }


Formula


For a relation R in set A


Reflexive


The relation is reflexive if (a , a) R for every a A


Symmetric


The relation is Symmetric if (a , b) R , then (b , a) R


Transitive


Relation is Transitive if (a , b) R & (b , c) R , then (a , c) R


Equivalence


If the relation is reflexive , symmetric and transitive , it is an equivalence relation.


Check for reflexive


Consider , (a, b) R (a, b)


(a, b) R (a, b) a + b = a + b


which is always true .


Therefore , R is reflexive ……. (1)


Check for symmetric


(a, b) R (c, d) a + d = b + c


(c, d) R (a, b) c + b = d + a


Both the equation are the same and therefore will always be true.


Therefore , R is symmetric ……. (2)


Check for transitive


(a, b) R (c, d) a + d = b + c


(c, d) R (e, f) c + f = d + e


On adding these both equations we get , a + f = b + e


Also,


(a, b) R (e, f) a + f = b + e


It will always be true


Therefore , R is transitive ……. (3)


Now , according to the equations (1) , (2) , (3)


Correct option will be (D)

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