I recently received a notification from WordPress that they had updated the template used for sending emails out to subscribers to WordPress blogs.

See https://wordpress.com/blog/2022/05/19/your-website-looks-great-so-should-your-emails/

I am curious to see how a bit of maths looks like in the new email template. So I should write something. (I’m subscribed to my own blog so I’ll see what the email looks like!)

I have a draft somewhere of something I was going to write about characteristic functions, counting and the inclusion/exclusion principle. Ah I see I already published Part 1 at

But I hadn’t had time to add the follow-up yet. So here is a sneak preview!

Another observation which can be helpful is the following. Suppose that and are elements of . Then , and this will be even **even **if and only if it is zero, i.e., if and only if . In terms of modular arithmetic, working modulo , for and in , we have

Here the forward implication is trivial, but the backward implication uses the restriction on and .

Since characteristic functions only take the values or , we have the following extension of the above. Let and be subsets of . Then (I must find a better way to write multiline LaTeX with alignment in WordPress! Perhaps they allow this by now and I just haven’t noticed?)

for all , we have

for all , we have

Here it is convenient to bring in some new terminology to save some writing.

Let and be integer-valued functions defined on $X$. We say that is congruent to modulo , and write if (and only if), for all , we have

In other words, we require to be **even** for all .

**To be continued!**