Mark the tick against the correct answer in the following: Let R be a

1.	Mark the tick against the correct answer in the following: Let R be a relation on N × N, defined by(a, b) R (c, d) ⇔ a + d = b + c. Then, R isA. 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 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, b) R (a, b) (a, b) R (a, b) ⇔ a + b = a + b which is always true . Therefore , R is reflexive ……. (1) Check for symmetric (a, b) R (c, d) ⇔ a + d = b + c (c, d) R (a, b) ⇔ c + b = d + a Both the equation are the same and therefore will always be true. Therefore , R is symmetric ……. (2) Check for transitive (a, b) R (c, d) ⇔ a + d = b + c (c, d) R (e, f) ⇔ c + f = d + e On adding these both equations we get , a + f = b + e Also, (a, b) R (e, f) ⇔ a + f = b + e ∴ It will always be true 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 R be a relation on N × N, defined by(a, b) R (c, d) ⇔ a + d = b + c. Then, R isA. 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

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, b) R (a, b)

(a, b) R (a, b) ⇔ a + b = a + b

which is always true .

Therefore , R is reflexive ……. (1)

Check for symmetric

(a, b) R (c, d) ⇔ a + d = b + c

(c, d) R (a, b) ⇔ c + b = d + a

Both the equation are the same and therefore will always be true.

Therefore , R is symmetric ……. (2)

Check for transitive

(a, b) R (c, d) ⇔ a + d = b + c

(c, d) R (e, f) ⇔ c + f = d + e

On adding these both equations we get , a + f = b + e

Also,

(a, b) R (e, f) ⇔ a + f = b + e

∴ It will always be true

Therefore , R is transitive ……. (3)

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

Correct option will be (D)