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 is
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)