Question

Let A be the set of all points in a plane and let O be the origin. Show that the relation $R=\left\{ \left( P,Q \right):P,Q\in A\text{ }and\text{ }OP=OQ \right\}$ is an equivalence relation.

Answer 1

Hint: In order to solve this question, we should know that an equivalent relation is nothing but a relation which is reflexive, symmetric and transitive relation. So, to prove relation R as an equivalent relation, we will prove it to be reflexive, symmetric, and transitive relation. We need to remember that for reflexive symmetry, we will prove $R=\left\{ \left( P,P \right):P,P\in A\text{ }and\text{ }OP=OP \right\}$.

Complete step-by-step answer:
In this question, we have been asked to prove that the given relation R is reflexive, symmetric, and transitive relation because we know that any relation is equivalent only when it is reflexive, symmetric, and transitive relation. Now, let us first go with reflexive relation. Reflexive relation is a relation that maps for itself. So, for $\left\{ \left( P,P \right):P\in A \right\}$, we have to prove that OP = OP. And we know that a line segment always has the same length and so OP will always be equal to OP.
So, we can say that relation r is reflexive relation.
Now, let us consider symmetric relations. Symmetric relation states that if $aRb$ then $bRa$ for a symmetric relation. Here we have been given the relation $R=\left\{ \left( P,Q \right):P,Q\in A\text{ }and\text{ }OP=OQ \right\}$. So, if it is true then we have to prove that $R=\left\{ \left( Q,P \right):P,Q\in A\text{ }and\text{ }OQ=OP \right\}$ is also true. Here we will consider the equality OP = OQ. We know that equality remains constant even if we interchange the sides of equality. So, we can write OQ = OP after interchanging the sides of OP = OQ, which is the result we wanted to obtain for $R=\left\{ \left( Q,P \right):P,Q\in A\text{ }and\text{ }OQ=OP \right\}$.
Hence, we can say that relation R is a symmetric relation.
Now, we will consider transitive relations. Transitive relation is a relation which states that if $aRb$ and $bRc$, then $aRc$. So, to check if the relation is transitive or not, we will consider P, Q and S. So, let us consider $PRQ,ORS$, that is OP = OQ and OQ = OS. Then we have to check whether $PRS$, that is OP = OS or not. We know that OP = OQ and we can also write it as,
OQ = OP ………… (i)
We also know that OQ = OS ………… (ii)
So, from equation (i) and (ii), we can say that OP = OS, which is the condition for $PRS$.
Hence, we can say that relation R is a transitive relation.
Therefore, we have proved that relation R is reflexive, symmetric, and transitive relation. Hence, relation R is equivalent relation.

Note: In this question, we can possibly make a mistake while proving relation R is symmetric relation by proving OP = PO, which would be wrong. In symmetric relation, we have to prove that if OP = OQ, then OQ = OP. Also, we should remember that whenever we are asked to prove any relation as an equivalent relation, we have to prove that it is a reflexive, symmetric and transitive relation.