# How to write down formally number of occurrences?

I want to write the following sentence formally:

The sequence $$S$$ contains elements of the set $$A$$. The probability value $$P(a)$$ for an element $$a$$ is defined as the number of its occurrences in the sequence $$S$$, divided by the count of all its elements.

I can write it the following manner:

$$S = (s_1, s_2, ..., s_n) : s_i \in A.$$ $$P(a) := {{ \left| \lbrace i \in \lbrace 1, 2, ..., n \rbrace : s_{i} = a \rbrace \right| } \over {n}}, \text{ given } n > 0\text{ and }a \in A.$$

It's, however, quite long and rather not elegant. Is there a simpler way to write this?

Edit:

There's always a solution, which involves breaking the formula to smaller parts:

$$\text{Let } C(x) = \left| \lbrace i \in \lbrace 1, 2, ..., n \rbrace : s_i = x \rbrace \right|.$$ $$P(a) := {C(a) \over n}.$$

It's more readable, but it's still not what I'm searching for...

1. If you are willing to use the "Iverson bracket notation", popularized by Knuth and others, you can say $$C(x) = \sum_{i=1}^n [s_i = x]$$

Here $[\ldots]$ are the Iverson brackets. $[P]$ is defined to be 1 if $P$ is true, and 0 if it is false.

2. People do sometimes use the Kronecker delta for this: $\delta_{ij}$ is defined to be 1 if $i=j$ and 0 if $i\ne j$, so you would have:

$$C(x) = \sum_{i=1}^n \delta_{xs_i}$$ or $$C(x) = \sum_{i=1}^n \delta(x, s_i)$$

but I think the Iverson bracket is more straightforward.

3. Most straightforward would be to write

Let $C(x)$ be the number of elements of $s_1,\ldots,s_n$ that are equal to $x$. Then…

The idea that this is somehow less "formal" than something involving a bunch of funny symbols is a common misapprehension.

• You know, I guess, that your third solution will suit my needs. After all, my priority is that someone understands my notation - writing the sentences as formally as possible is not really as important. Commented Mar 11, 2013 at 7:46
• Agreed. I think the third solution provided is optimal for readability. A very classical approach. Commented Mar 9, 2016 at 8:07
• I just learned about Iverson brackets, nice Commented Oct 1, 2019 at 11:32

Using the number symbol "#",

$$\DeclareMathOperator*{\countif}{\#} P(a) := \dfrac{\countif\limits_{i=1}^{n} (s_i=a)}{n}, \text{ given } n > 0\text{ and }a \in A.$$

It's also not quite correct. How about $|\{i\in\{1,\ldots,n\}\colon s_i=a\}|$ in the numerator? If you view the sequence $S$ as a function $S\colon\{1,\ldots,n\}\to A$, $i\mapsto s_i$, then you might even write $|S^{-1}(a)|$ for the numerator. And the denominator should simply be $n$ (which you implicitly defined for $S$). As $S$ is not (primarily) a set, $|S|$ looks strange.

• You're right about my notation, I corrected it. However, I don't like the solution with $S^{-1}$ much, that looks like redefining the sequence to simplify the notation. I thought rather of a mathematical construct corresponding to a 'count of' function. Isn't there one? Commented Mar 10, 2013 at 15:55
• @Spook Actually, that normally isn't really redefining the ntotion of sequence. But nevertheless, isn't $|\{\ldots\}|$ what corresponds to a count function after all? All that matters is that you can define your frequency notion clear and unambiguously, not necessarily with less then five symbols ... Commented Mar 11, 2013 at 7:41

2 ideas:

1) Using the hashtag or number symbol "#", e.g:

$$\#(a \in S)$$

It is fairly common within literature, see: 2008. Elements of Statistical Learning 2nd Ed, Chapter 9.2.2. Hastie, Tibshirani, Friedman

2) Using the indicator function (returns 1 if condition true, else 0) in a sum, e.g: $$\sum\limits_{i=1}^{n}{\textbf{1}_{a_i \in S}}$$

or

$$\sum\limits_{i=1}^{n}{I({a_i \in S})}$$

It is probably the most common and 'mathematical', see: 2008. Elements of Statistical Learning 2nd Ed, Chapter 9.2.3. Hastie, Tibshirani, Friedman

Usage example for original post:

$$\mathbb{P}(a) = \frac{\sum\limits_{i=1}^{n}{\textbf{1}_{a_i \in S}}}{n} \forall \ a \in S \in A$$ where

$$n$$ = number of elements in S

$$\textbf{1}_a$$ = indicator function that returns 1 if the element is a

• I suppose, that #(...) is so far most clear notation for me. The Iverson bracket [...] seems relatively clear too, but I guess, the hash-notation is more obvious. I guess, that it would be beneficial if matematicians agreed on some common notation at some point, because - especially in computer science - it is used very often. Commented Sep 28, 2020 at 7:37