Identify the equivalence relation on a set

Question

Let $A$ be the set of all points in the $x - y$ plane. Define a relation $\sim$ on A as follows:

For $a, b\in A$, $a\sim b$ if and only if $a$ and $b$ have the same distance from the origin. Is $\sim$ an equivalence relation on $A$?

Answer 1

Yes, this is an equivalence relation, because it satisfies

• reflexivity: a point $a \in A$ has the same distance from the origin as itself;
• symmetry: if $a$ and $b$ have the same distance from the origin, so do $b$ and $a$.
• transitivity: if $a$ and $b$ have the same distance from the origin and $b$ and $c$ have the same distance from the origin, then $a$ and $c$ have the same distance from the origin.