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.
$\Box$
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.
$\Box$
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.
$\Box$
$\sim$ has been shown to be reflexive, symmetric and transitive.
Hence by definition it is an equivalence relation.
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Exercise $7$