Equivalence Relation/Examples/Points on Same Line

Example of Equivalence Relation
Let $P$ be the set of points in the plane.

Let $\sim$ be the relation on $P$ defined as:
 * $\forall x, y \in P: x \sim y \iff \text {$x$ and $y$}$ both lie on the same horizontal line

Then $\sim$ is an equivalence relation.

Proof
In the Cartesian plane, the characteristic of a horizontal line is that the $y$ coordinates of all its points are equal.

Checking in turn each of the criteria for equivalence:

Reflexivity
Let $p = \tuple {x, y} \in P$.

Then $p$ is trivially on the same horizontal line as itself.

Thus $\sim$ is seen to be reflexive.

Symmetry
Let $p_1 = \tuple {x_1, y_1}, p_2 = \tuple {x_2, y_2} \in P$.

Let $p_1 \sim p_2$.

Then $y_1 = y_2$, and so $y_2 = y_1$.

Hence $p_2 \sim p_1$

Thus $\sim$ is seen to be symmetric.

Transitivity
Let $p_1 = \tuple {x_1, y_1}, p_2 = \tuple {x_2, y_2}, p_3 = \tuple {x_3, y_3} \in P$.

Let $p_1 \sim p_2$.

Let $p_2 \sim p_3$.

Then $y_1 = y_2$ and $y_2 = y_3$.

Hence $y_1 = y_3$ and so $p_1 \sim p_3$.

Thus $\sim$ is seen to be transitive.

$\sim$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.