Equivalence Relation/Examples/Months that Start on the Same Day of the Week

Example of Equivalence Relation
Let $M$ be the set of months of the year according to the (usual) Gregorian calendar.

Let $\sim$ be the relation on $M$ defined as:
 * $\forall x, y \in M: x \sim y \iff \text {$x$ and $y$ both start on the same day of the week}$

Then $\sim$ is an equivalence relation.

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity
Let $x \in M$.

Then $x$ starts on the same day of the week as itself.

Thus $\sim$ is seen to be reflexive.

Symmetry
Let $x, y \in M$.

If $x$ starts on the same day of the week as $y$, then $y$ starts on the same day of the week as $x$.

Thus $\sim$ is seen to be symmetric.

Transitivity
Let $x, y, z \in M$.

Let $x$ start on the same day of the week as $y$.

Let $y$ start on the same day of the week as $z$.

Then $x$ starts on the same day of the week as $z$

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.