Q. 85.0( 1 Vote )

Mark the tick against the correct answer in the following:

Let Z be the set of all integers and let R be a relation on Z defined by a R b a≥ b. Then, R is

A. symmetric and transitive but not reflexive

B. reflexive and symmetric but not transitive

C. reflexive and transitive but not symmetric

D. an equivalence relation

Answer :

According to the question ,


Given set Z = {1, 2, 3 ,4 …..}


And R = {(a, b) : a,b Z and a ≥ b}


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,a) (b,b)


a a and b b which is always true.


Therefore , R is reflexive ……. (1)


Check for symmetric


a R b a b


b R a b a


Both cannot be true.


Ex _ If a=2 and b=1


2 1 is true but 1 2 which is false.


Therefore , R is not symmetric ……. (2)


Check for transitive


a R b a b


b R c b c


a c


Ex _a=5 , b=4 and c=2


5≥4 , 4≥2 and hence 5≥2


Therefore , R is transitive ……. (3)


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


Correct option will be (C)

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