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 but neither symmetric nor transitive B. symmetric but neither reflexive nor transitive C. transitive but neither reflexive nor symmetric D. an equivalence relation
Answer» Correct Answer is (B) symmetric but neither reflexive nor transitive Given set S = {x, y, z} And R = {(x, y), (y, z), (x, z) , (y, x), (z, y), (z, x)} 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 not reflexive ……. (1) Check for symmetric Since , (x,y) ∈ R and (y,x) ∈ R (z,y) ∈ R and (y,z) ∈ R (x,z) ∈ R and (z,x) ∈ R Therefore , R is symmetric ……. (2) Check for transitive Here , (x,y) ∈ R and (y,x) ∈ R but (x,x) ∉ R Therefore , R is not transitive ……. (3) Now , according to the equations (1) , (2) , (3) Correct option will be (B)

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 but neither symmetric nor transitive B. symmetric but neither reflexive nor transitive C. transitive but neither reflexive nor symmetric D. an equivalence relation

Answer»

Correct Answer is (B) symmetric but neither reflexive nor transitive

Given set S = {x, y, z}

And R = {(x, y), (y, z), (x, z) , (y, x), (z, y), (z, x)}

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 not reflexive ……. (1)

Check for symmetric

Since , (x,y) ∈ R and (y,x) ∈ R

(z,y) ∈ R and (y,z) ∈ R

(x,z) ∈ R and (z,x) ∈ R

Therefore , R is symmetric ……. (2)

Check for transitive

Here , (x,y) ∈ R and (y,x) ∈ R but (x,x) ∉ R

Therefore , R is not transitive ……. (3)

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

Correct option will be (B)