Mark the tick against the correct answer in the following: Let S be t

1.	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 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» Correct Answer is (D) an equivalence relation 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)

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 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»

Correct Answer is (D) an equivalence relation

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)