Q. 65.0( 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) is divisible by 3. Then, R isA. reflexive and symmetric but not transitiveB. reflexive and transitive but not symmetricC. symmetric and transitive but not reflexiveD. an equivalence relation

According to the question ,

Given set Z = {1, 2, 3 ,4 …..}

And R = {(a, b) : a,b ∈ Z and (a-b) is divisible by 3}

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)

(a - a) = 0 which is divisible by 3

(a,a) R where a Z

Therefore , R is reflexive ……. (1)

Check for symmetric

Consider , (a,b) R

(a - b) which is divisible by 3

- (a - b) which is divisible by 3

(since if 6 is divisible by 3 then -6 will also be divisible by 3)

(b - a) which is divisible by 3 (b,a) R

For any (a,b) R ; (b,a) R

Therefore , R is symmetric ……. (2)

Check for transitive

Consider , (a,b) R and (b,c) R

(a - b) which is divisible by 3

and (b - c) which is divisible by 3

[ (a-b)+(b-c) ] is divisible by 3 ] (if 6 is divisible by 3 and 9 is divisible by 3 then 6+9 will also be divisible by 3)

(a - c) which is divisible by 3 (a,c) R

Therefore (a,b) R and (b,c) R then (a,c) R

Therefore , R is transitive ……. (3)

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

Correct option will be (D)

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