Mark the tick against the correct answer in the following:
Let S be the set of all straight lines in a plane. Let R be a relation on S defined by a R b ⇔ a || b. Then, R is
According to the question ,
Given set S = {x, y, z}
And R = {(x, x), (y, y), (z, z)}
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
Since , (x,x) ∈ R , (y,y) ∈ R , (z,z) ∈ R
Therefore , R is reflexive ……. (1)
Check for symmetric
Since , (x,x) ∈ R and (x,x) ∈ R
(y,y) ∈ R and (y,y) ∈ R
(z,z) ∈ R and (z,z) ∈ R
Therefore , R is symmetric ……. (2)
Check for transitive
Here , (x,x) ∈ R and (y,y) ∈ R and (z,z) ∈ R
Therefore , R is transitive ……. (3)
Now , according to the equations (1) , (2) , (3)
Correct option will be (D)