To show that the relation defined on the set A of all points in a plane (where O is the origin) is an equivalence relation, we need to verify that it satisfies three properties: reflexivity, symmetry, and transitivity.
1. Reflexivity:
A relation is reflexive if every element is related to itself. For a point P in A, we need to show that P is related to P. Since OP = OP (the distance from the origin O to point P is equal to itself), the reflexivity condition holds.
2. Symmetry:
A relation is symmetric if for any two points P and Q, if P is related to Q, then Q is related to P. If we have that OP = OQ, it implies a certain relationship between P and Q. If OP = OQ, then by the nature of equality, we also have OQ = OP, proving that Q is related to P as well. Hence, the symmetry condition is satisfied.
3. Transitivity:
A relation is transitive if whenever P is related to Q and Q is related to R, then P is also related to R. So, if OP = OQ and OQ = OR, it follows from equality that OP = OR. Therefore, P is related to R, confirming that the transitivity condition holds.
Having shown that the relation satisfies reflexivity, symmetry, and transitivity, we conclude that the relation defined on the set A of all points in a plane is indeed an equivalence relation.