Q. 155.0( 1 Vote )

# Mark the tick against the correct answer in the following:Let A be the set of all points in a plane and let O be the origin. Let R = {(P, Q) : OP = QQ}. Then, R isA. reflexive and symmetric but not transitiveB. reflexive and transitive but not symmetricC. symmetric and transitive but not reflexiveD. an equivalence relationThere is printing mistake in the question…R should be R = {(P, Q) : OP = OQ}Instead of R = {(P, Q) : OP = QQ}

According to the question ,

O is the origin

R = {(P, Q) : OP = OQ }

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 , (P , P) R OP = OP

which is always true .

Therefore , R is reflexive ……. (1)

Check for symmetric

(P , Q) R OP = OQ

(Q , P) R OQ = OP

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

Therefore , R is symmetric ……. (2)

Check for transitive

(P , Q) R OP = OQ

(Q , R) R OQ = OR

On adding these both equations, we get , OP = OR

Also,

(P , R) R OP = OR

It will always be true

Therefore , R is transitive ……. (3)

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

Correct option will be (D)

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