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

Solution:

A relation is a subset of a cross-product of two sets. If A and B are two sets, then a relation from A to B is a subset of A × B.

Let O be the origin and let P, Q, X be any three points in a plane.

Let us denote this relation by R = {(P, Q): OP = OQ} for O is the origin.

Now (P, P) ∈ R since OP = OP for any point P.

So the relation is a reflexive relation.

Again this relation is symmetric as if (P, Q) ∈ R ⇒ (Q, P) ∈ R

Since OP = OP ⇒ OQ = OP for all P, Q.

Also, this relation is transitive as if (P, Q) ∈ R, (Q, S) ∈ R ⇒ (P, S) ∈ R

Since OP = OQ, OQ = OS ⇒ OP = OS for all P, Q, S.

Thus an equivalence relation on a set A is defined as a subset of its cross-product, that is, A × A. Equalities is an example of an equivalence relation.The relation R = {(P, Q): OP = OQ} is an equivalence relation.

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

Summary:

If 'a' be the set of all points in a plane and 'O' be the origin then the relation R = {(P, Q): OP = OQ} is an equivalence relation.