To demonstrate that the relation for points in a plane relative to the origin is an equivalence relation, we must establish that it satisfies three essential properties: reflexivity, symmetry, and transitivity.
1. Reflexivity
A relation is reflexive if every element is related to itself. In this context, for any point p in the set A, we need to show that p is related to p. This can be expressed mathematically as:
For every p in A, p is related to p (denoted as p ~ p).
Using the definition of the relation, in terms of the origin o, we see that the distance from o to p is always equal to itself, confirming reflexivity.
2. Symmetry
A relation is symmetric if for any two points p and q, whenever p is related to q, q is also related to p. Mathematically:
If p ~ q (meaning the distance from o to p is the same as from o to q), then q ~ p.
This property holds true since if the distances are equal (say, both distance equals d), then it naturally follows that the distance from o to q is also d. Thus, the relation is symmetric.
3. Transitivity
A relation is transitive if whenever p is related to q and q is related to r, then p must also be related to r. Mathematically:
If p ~ q and q ~ r, then p ~ r.
From our relation, if the distances from o to p and q are equal (both are d1) and the distances from o to q and r are equal (both are d2), then for p and q to be related, d1 must equal d2. Hence, p ~ r must also hold if the distances are equal, fulfilling the transitive property.
Conclusion
Having verified that the relation among points based on their distances to the origin is reflexive, symmetric, and transitive, we can confidently conclude that the relation is indeed an equivalence relation. This property highlights a fundamental way in which points in a plane, viewed in relation to an origin, can be categorized into equivalence classes based on their distances.