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

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Examples of Equivalence Class

Let $M$ be the set of months of the (calendar) 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}$


The set of equivalence classes under $\sim$ depends on whether the year is a leap year.

For a non-leap year, the set of equivalence classes is:

$\set {\set {\text {January}, \text {October} }, \set {\text {February}, \text {March}, \text {November} }, \set {\text {April}, \text {July} }, \set {\text {May} }, \set {\text {June} }, \set {\text {August} }, \set {\text {September}, \text {December} } }$


For a leap year, the set of equivalence classes is:

$\set {\set {\text {January}, \text {April}, \text {July} }, \set {\text {February}, \text {August} }, \set {\text {March}, \text {November} }, \set {\text {May} }, \set {\text {June} }, \set {\text {September}, \text {December} }, \set {\text {October} } }$


Proof

We have that:

The months with $30$ days are:
$\text {April}, \text {June}, \text {September}, \text {November}$
The months with $31$ days are:
$\text {January}, \text {March}, \text {May}, \text {July}, \text {August}, \text {October}, \text {December}$
In a non-leap year, $\text {February}$ has $28$ days
In a leap year, $\text {February}$ has $29$ days.


Let month $m$ have $m_d$ days in it.

Let month $m$ start on day $d$, where $d$ is in the range $0$ to $6$ (which day of the week corresponds to which number is irrelevant at this stage).

Then month $m + 1$ starts on day $\paren {d + m_d} \pmod 7$.


For reference:

\(\ds 28\) \(\equiv\) \(\ds 0 \pmod 7\)
\(\ds 29\) \(\equiv\) \(\ds 1 \pmod 7\)
\(\ds 30\) \(\equiv\) \(\ds 2 \pmod 7\)
\(\ds 31\) \(\equiv\) \(\ds 3 \pmod 7\)


Without loss of generality, let $\text {January}$ start on day $0$.

Then the sequence of the days which are the $1$st of the month is as follows:


For a non-leap year:

$\tuple {0, 3, 3, 6, 1, 4, 6, 2, 5, 0, 3, 5}$

This sequence is A189915 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


For a leap year:

$\tuple {0, 3, 4, 0, 2, 5, 0, 3, 6, 1, 4, 6}$

This sequence is A189916 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The result follows.

$\blacksquare$


Sources